1999 Mercury Cougar engine starts and won't stay running

Hello, for some reason, when I start my car, it starts, rpm goes up and then car just dies. After a few tries and putting my foot on the gas, to prevent it from stalling, the car will stay started, but it is struggling. The rpm's go up then down, up and down. The problem get's a little better the longer I run the car. I recently had the car in the shop and was told that the alternator was bad, so I had it replaced. The car ran good for about a week and now the problems are back. Any advise would be greatly appreciated.