Who has final say in the duration of the offender's prison stay in a determinate sentence?

A "Judge" has final say in the duration of the offender's prison stay in a determinate sentence.

A judge is a man who directs court procedures, either alone or as a part of a board of judges. The forces, capacities, strategy for arrangement, discipline, and preparing of judges change generally crosswise over various jurisdictions. The judge should lead the trial in an unbiased way and, commonly, in an open court.

I believe the answer is: Judge

Even though juries have the power to determine whether a defendant is guilty or not, they do not have a final decision in determining duration of prison stay.

That power (along with the power to maintain order during the trial) fall at the hands of the judges.