It is narrated from Sayyidah Aishah (may Allah be pleased with her) that the Messenger of Allah (peace and blessings be upon him) would perform the ritual bath, then pray the two rak'ahs of the morning (Fajr), and I do not think that he (peace and blessings be upon him) performed a fresh ablution after the bath.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 851