1. Better breath support.
2. Bright eyes and raised eyebrows.
3. Rehearse with pedal drones, then without, then with, then without...
4. Solfege the entire piece many times.
5. During rehearsal, play the final note on the organ at the end of each phrase with the singers.
6. Try learning the piece without a keyboard, so as to avoid tempered-tuning problems.
7. Practice singing non-tempered scales and intervals (especially 5ths). (If you don't already know what that sounds like, or need help- many synthesizer keyboards allow you to switch the tuning system. Also, there are some online tools to help.)
8. Change your own and your choir's free-time listening habits to include more unaccompanied chant recordings and early polyphony.

And finally...

9. Unless you need a direct attack into another piece in the same "key," try not to worry about it too much, as most people won't really notice.

Just FYI, Ted Marier has an article on this very subject that appeared in Sacred Music Journal.

We do a piece at Christmas that has SATB refrain and chanted verses. All without organ. We had the same problem during the verses where the pitch would just fall and fall every time. Then, when we got to the refrain, no one could find their notes because of the dip in pitch. I ended up employing an organ pedal point for the verses...low pedal note that I held through the whole verse. It worked out very well in terms of keeping the pitch up and it sounded really good, too. This works for music with a tonal center, of course. Not sure how it would work with modal melodies.

The last parish choir I sang in did all of its rehearsals without keyboard except to give the starting "doh" pitch. We had to mentally find our starting pitch from that (whether in major or minor key) and sing everything in tonic solfa (solfege) syllables. This did more than anything else to improve our pitch. We still weren't perfect, but much better than before when we relied on the keyboard to keep us going.

Before you sing with the text, sing it with the solfegge syllables. Give it a shot. Watch the half steps, they are the usual culprit.

What I will do is find out where the pitch falls and listen and rehearse this spot, then the next and the next (the drone will help this). Yes, sitting up straight, using the diaphragm, and general good technique all help. The problem in my choir is that I have a couple of voices who are incessantly going flat. It is generally just these two women and they refuse to change. It is very frustrating indeed!

In the end I have raised and taught my boys, who have developed perfect pitch and they have been keeping the entire soprano section from going flat in my choir. Honestly, for 11 and 13 year olds, they have really been a blessing! If you can get a couple of these, do!

Pitch is not a function of breath..., yet all speaking and singing is fundamentally reliant on the breath. Without air there is no pitch. Pressing phonation can cause many problems including vocal nodes and tension.
In many cases the sag in pitch (a perfect 5 or even a major third leap) results from not enough breath support to maintain pitch. It becomes a problem of focus and resonance. Speak through a phrase with energy and then immediately sing it quite freely. The amount of energy (or breath support) used to speak an energetic phrase is often what is required to sing. Remember singing is very much like speaking.

In the case of a whole choir losing pitch, there may be a variety of reasons for it.
One is conflicting ideas of the vowel sound. Two different sung versions of the vowel can cause confusion to the ear and the result is often flat. Very basic vowel exercises (ie. ma, me, mi, mo, mu) will help to unify the choir's concept the pure Italianate vowels. The director must then ensure through the exercise that people are listening whilst singing and that goal of the exercise is being met and not just blindly sung.

Many voice teachers will give differing opinions but the remedies often yield similar results. Usually because a correct amount of focus from the singer on what he is actually doing when he sings starts to correct some of these problems.

Inevitably in a choral situation each singer will have varying levels of vocal technique. So while it's true that there are some "tricks" such as raising the eyebrows which seem to benefit the composite group sound overall, it's also true that it is not solid vocal technique to rely on such "crutches." "Raising the eyebrows", or "smiling with the eyes", "smelling a rose", "thinking of the beginning of a laugh, cry, sneeze or yawn" are all "tricks" to help focus the sound in the mask, and while each has it's value, it can also cause other technique problems such as creating unnecessary tension elsewhere.

Of course the reality is, outside of a professional choir, each singer won't have the benefit of private lessons, nor is there time in a choral rehearsal to thoroughly teach proper technique. Therefore such tricks, when used judiciously, can be useful. It would be better to "think" of raising the eyebrows (or any of the others mentioned above) rather than contorting the face unnaturally.

Likewise with support. "Supporting" the breath is obviously very important, but telling an untrained singer to "support more" is not always helpful since it can be interpreted differently. Over-supporting (hypertension) is just as bad as under-supporting (hypotension).

In my humble opinion, a cappella singing is an ideal that can't be reached by the average volunteer choir of untrained singers--especially in a parish setting where there is only one rehearsal per week and the repertoire is constantly changing. A concert choir who has more time to prepare a piece is another story. I would rather have my choir sing good quality music with an organ or other instruments doubling the parts than sing it poorly without any accompaniment or not be exposed to that music at all because it is too challenging.

In my experience, the problem is almost always in the descending 1/2 step.

Before you try to fix this problem, you might try moving DO drastically down or up to a point where they are comfortable, which may solve the problem. You may be surprised how far you have to move DO to find a common tessitura in your group at which they are all confortable.

C is not DO - but you knew that already, right? ;<)

What a great bunch of ideas! With my schola, one thing I tried was to simply see where we gravitated towards, and then start in that key. It often worked.

I also tried putting the stronger singers a bit behind the others, and that helped too.

My itchy finger was provoked by only one line of an otherwise thoughtful post and I assure EarlGrey that the potshot was meant in a friendly spirit as well. Was it really that hard to say "not always"? ;-)

James Jordan recommended concentrating on listening through your right ear. This is actually impossible to do (listening through one ear and not the other, unless you have hearing impairment), but simply trying to do so helps pitch problems.

Also, vowels. Non-uniformity of vowels can make things sound out of tune even if they aren't.

Jonathan, the link, although it reads correctly, points to this page. Here is the correct PDF link.