1993 Mazda Protege Battery dies after charging

I have a 1993 Mazda Protege and it will not keep a charge. I charge the battery and it will start up just fine but after 2 to 5 miles it will die and I have to charge it again. I tested the alternator 3 times and every time a different place says it is fine. For some reason it is not charging the battery like it should. I also disconnected the battery cable from the battery while the car was on and it died automatically. Please help I bought this car as a surprise for my daughters graduation. Thanks.