I've never heard of any Roman song recorded in its entirety.

However, we do know of some fragments. The first that springs to mind is one that you're probably familiar with - part of the song sung at Caesar's triumph over Gaul. According to Suetonius,

Home we bring our bald whoremonger; Romans, lock your wives away! All the bags of gold you lent him Went his Gallic tarts to pay.

Again, I'm unaware of anything but stray verses like the above, but I'll also concede that Roman music is quite outside my expertise and I hope someone can provide you a more complete answer.