As supercharged mentioned, have you or can you check fuel pressure? Is the check engine light on or was it on before this started happening? I am not talking about while you are trying to start it or when it dies, many light come on at that point.
By the fact it ran fine when you put a new battery in, for awhile and then did it again, I am thinking you have charging issues and it is the alternator. Can you pull the alt out and have it checked?
You also said the old battery would not hold a charge, thus you repalced it. It should have if it was in good shape, but again, thus tells me is was killed beyond its useful life and damaged.
When alternators go bad, the car runs off the battery until it kills it too...
