Battery not staying charged

I have a 2001 Tiburon with 77,000 miles on it. After returning from a trip to south Texas (roughly 700 miles without a hitch), I noticed my dash lights/headlights dimming and shortly thereafter my car began to idle roughly and sputtering until it died. After letting the car sit for roughly an hour, it started up and ran fine until about 5 minutes later when it acted exactly as before (dash lights dimming, losing all electrical power slowly until it died again). I removed the alternator and battery, and had them tested at AutoZone, where I was told the alternator was good but the battery was bad. I went ahead and got a new alternator and battery and just finished installing them both (after I charged the battery). The car did the exact same thing as before. I'm at a loss as to what could be causing this since It's run fine for the 4 years I've had it and I've never had any problems with the electrical system until now. Any advice would be helpful, Thank you. -Travis