Its not a pass fail test, but it really does depend on the person taking it and the score they are looking for. I think a month is plenty of time for a lot of people, especially if you can devote a lot of time to it. I know personally I have been "prepping" for the GMAT for about a month and a half but I study a couple times a week for an hour or two a time at most. So someone who has a the time to focus more energy could easily do more in a week than I have done in a month. I am hoping that over the next month or two I can focus more time in, if I could devote an hour or two a night and a few extra on weekends for practice tests I would feel a month was adequate.
What is the average GMAT at the school? If the rest of your profile is strong enough to get you in and its not a super competitive program then you should be able to reach the average in a month.