because the alternator is new means nothing. using a voltmeter, check the voltage & ensure that it is close to 13.8 volts while running. if it is staying at or near there, how hot is it getting? you haven't said how long it sits between startings..... if it's running & dies, you can't start it right away with its own battery? sounds like it's not charging to me if that's the case - again, voltmeter & check. if it's showing 12v or less while running, your alternator is NOT working or you have a major short somewhere (not likely though). an easy check to ensure alternator is working if you don't have a voltmeter, start the car, & carefully remove one battery cable. if the car dies, the alternator isn't working, hooked up wrong, whatever - but it's not putting out any power.