Dead Battery after 20 minutes. Alternator?

I went to get in my car the other day to find out that the autolock was dead and the electronic on the dash were dead and of course not even a click when starting it. I got a jump and it started right up. Knowing the battery was very old (maybe the original) I went to the store for a new one. After driving it for 20 minutes to get to the store I tried restarting to see if the alternator had recharged it and it did, so another confirmation it was the battery. So I swapped it out and all was good. I drove it straight home and parked it. When I went to start my car the next day and it was completely dead like before. My next thought is the alternator but it doesn't really sound like a bad alternator. Plus when does a new battery go completely dead after 20 minutes? Any thoughts?

Thx!