Island travel 1 - Since you are on the same airline for both your inbound and outbound flight a 1 hour connection time is doable. AA has sevearl terminals at MIA but they are all linked by a passageway that is inside the secure area. If memory serves me correct it takes less than 20 minutes to walk from the two furthest apart gates. Chances are your gates will be closer so the walk time will be less. Assuming your inbound flight is relatively on time (coming from Atlanta weather shouldn't be a problem) both you and your checked luggage should have no trouble making the connection. Figure if your plane arrives on-time, it will take 10 minutes to taxi and arrive at the gate and it will be another 10 to 15 minutes before you deplane. That leaves you 35 to 40 minutes for you to get to your outbound gate. If it takes you 10 minutes to walk between gates you still have at lease a 25 minute cushion.
For what it is worth, sometimes if an inbound flight is delayed and there are a fair number of passengers on that flight making the same connection AA will "hold" the out-bound flight, however, you can not rely on this happening especially if you are the only connecting passenger (there's no way for you to tell ahead of time). You'd be wise to find out when the next flight to PLS departs so you know what your alternatives are should you encounter a problem.
FYI - I've made many a connection with a 1 hour (or slightly less time) connection but I've also missed my share. You have to take into consideration Murphy's Law as it applies to airplane travel which simply states that the shorter your connection time is the more your chances of encountering any number of delays increase.
As I said, 1 hour is doable but if you can get something closer to 90 minutes that would be better.