What is the ruling on praying Fajr between the first and second calls to prayer, and sufficing with it without repeating it after the second call to prayer?

The entry of the time (for prayer) is a condition for the validity of the prayer. So, whoever prays the obligatory prayer before its time, their prayer is not valid as an obligatory prayer. It is only valid as a supererogatory prayer. If it becomes clear that the prayer was performed before its due time, then it is obligatory to make up for it because it is a debt upon one's conscience. One must also repent to Allah for this grave error.