I play guitar & sing in a rock 3-pieceso have to do this for whole gigs at a time - It's possibly a little different doing this on guitar to piano, but here's how I've managed it:
1) Assuming you can play the chords already, concentrate on getting the singing part right (as Marian suggests). get the rhythm of the syllables and the notes etc all sorted out so that you're comfortable with this. As you say, if anything 'lapses', you really don't want it to be the vocals so I'd suggest making this drive things.
2) On a guitar, if it's a "strummy" kind of song like "Come up & see me" by Steve Harley & the Cockney Rebel, the rhytm is simple. On a guitar it translates to waving your arm up & down as you strum. That acts as a kind of metronomic 'pulse' which a part of your body is setting up. (more on this later)
3) Learn how the syllables fit with this pulse/beat- sometimes that might mean slowing it right down and going over which syllable the beat occurs. For example the"Smile" of "come up & see me make me smile" is sometimes right on the beat, sometimes (later in the song) he delays it just a little.
4) As I mentioned, on a guitar the strumming gives a pulse if it's a simple rhythm, but funkier tunes like. . umm .. "Play that funk music white boy" it's a bit more choppy. Assuming it's a repeating riff, I've found a good way is to get very comfortable with playing the riff repeately so that you can kind of "set it going" while you use some other part of your brain to start singing the lyrics. this is also where looking closely at where the syllables fit with the beat (3) becomes important.
Sometimes I tap my foot or something to get that 'pulse'. On a piano, I'm not sure how you produce a rhytm like that but if you need the pulsing thing (f'nar again), then tapping your foot in time should do it.
I hope this helps- it's definitley what's worked for me.