The battery has been replaced but the battery still loses power and runs down once its turned off

will it turn over after it stalls? if alternator is not charging the battery will go dead while driving, if thats not your problem you need to ck for spark and fuel, a bad fuel pump can cause that problem, runs when cold then loses pressure once its run for a while

change alternator

Not sure I follow your story.
Have you checked the output of the alternator while it is on the engine, and is it between 13 and 14 volts ? Does the battery lose power while the engine is running or after you turn the engine off ?
Is the alternator output normal but not reaching the battery ?

was the key stuck in the ignition before you change the battery? Is it the right key or it a fresh cut key

Ck your alternator output, should be between 13.8 and 14.2 volts, A bad alt will cause the battery to discharge when driving, Once voltage drops to a certain level (9.8 I think ) the ecm starts to shut down causing all kinds of strange problems, like running bad ,trans not shifting right, even the radio ! It happened to me in my 95 chev p/u

have the battery load tested
clean the terminals and battery posts and tighten
check for good clean tight ground point for the battery

just simply replace your belt then buy another one with the same size or model of your belt..

Bad Alternator !

From what you describe, it sounds like your alternator died. The battery's main function is to provide power to start the vehicle. Once started, the alternator takes over and provides the main source of power to the truck, and also recharges the battery.
When the alternator stops working, the car starts to run solely off the battery. It will work for a little while, but as the battery starts to lose power, strange things will happen with the electronics. The computers and other electronics that control the vehicle are very dependent on a constant, steady flow of voltage. When that voltage dips below 11 or 10 volts, the car may start to run badly or strange things may happen with electrical system.
When there was finally not enough power to keep the car running, it shut off. Of course, once this happens, there will not be enough power left to start the truck back up.

I would check my wiring harness for faulty connections and possibly burnt wires. Also trace the wires from your alternator to the battery. check them over. The battery may need to be replaced. If you let it run down alot and kept jumping it.