Chev Venture van 2002, car battery is only 2 months old.

1. Turning the ignition swich makes no sound, no dash lights and
stereo is dead too.

2. Connect jumper cable from another car battery and it starts
instantly.  Remove jumper, engine keeps running, drive on highway for
20 minutes.

3. Turn off ignition and engine.

4. Turn on ignition, but will not start exactly like in step 1.
Connect jumper and it starts and runs like in step 2.

Please let me know what are the most probable cause of starting
problem is.  Is it the battery ($70-100 + 1hour labour) or alternator
($200-400 + 2 hour labour) not being able to recharge to battery.
Want to prepare myself before I decide to go to a workshop.

Thanks in advance to all who reply.