I recently had to replace a game disc of mine that was broken. I have not played it yet. Will my saved game fro, the first disc transfer over to the second disc?
You don't need to worry about it. You will be able to access your save game as usual.
Your save data will not transfer from one disk to another, because the disks are not writeable. The save data are located on your HDD (or if you don't have an HDD, they are located on your memory card or USB drive).
This means that as long as your HDD doesn't break, your save game will remain safe, even if you have to change to a new disk (as long as it's for the same game) or decide to buy the digital version instead.
Make sure that the disk is for the same region too, as different regions may use incompatible save data.