Muhammad reported on the authority of some of his teachers that Ubayy b. Ka'b led them in prayer during Ramadan. He used to recite the supplication (in the witr) during the second half of Ramadan.
أَنَّ أُبَىَّ بْنَ كَعْبٍ أَمَّهُمْ يَعْنِي فِي رَمَضَانَ وَكَانَ يَقْنُتُ فِي النِّصْفِ الآخِرِ مِنْ رَمَضَانَ