He did not know that the month had started until after Fajr time began

Question :

What is the ruling on the fast of the person who did not know that the month of Ramadhan had begun until after the time of Fajr had entered, due to sleep or other than that?

Answer:

Whoever did not know that the month of Ramadhan had begun until after the time of Fajr had entered, then he must refrain from those things that break the fast for the rest of his day, because it is a day of Ramadhan. It is not permissible for the person who is a resident of a place and healthy to partake of any of the things that would break the fast during Ramadhan. He must also make up for the day of fasting because he did not pass the night with the intention to fast before the time of Fajr. It has been confirmed from the Prophet that he said:
"Whoever does not pass the night with the intention to fast before the (time of) Fajr, then he has no fast."

The great scholar Ibn Qudamah reported this in Al-Mughni and it is the opinion of most of the Fiqh scholars. What is intended by this is the obligatory fast, and this is due to the noble Hadith which we have mentioned. In reference to the voluntary fast, it is permissible to begin it during the daytime as long as the person has not partaken of anything that would break the fast. This is because that which proves this has been authenticated from the Prophet. We ask Allâh to help the Muslims to do that which pleases Him and that He accept from them their fasting and standing in prayer during the night. Verily, He is All- Hearing, Most Near.

Source:
Ash-Shaykh Ibn Baz
Fatawa Islamiyah, Vol. 3 Pages 239-240