What could be the problem if your car will not start after running on a brand new battery after a month?

The alternator probably isn't putting out properly. Go to AutoZone and have them check the alternator. If the new battery went completely dead, and you know the alternator is good, the culprit could be a "slow draw". This would be caused by a trunk light that stays on when the trunk is closed, a vanity mirror light, glove compartment light, console light; something that you assume goes out when it's closed, but doesn't. Check all of your courtesy light switches in these places to make sure they make proper contact. A glove compartment or console door that doesn't close properly could be the culprit. Until you find the problem, disconnect your battery when you're in for the night! If you can't resolve it on your own, find a shop that has a low amps probe. It'll determine where the draw is coming from.