Assuming you have a fairly well designed database, and a reasonably capable server, you should be able to manage a million rows in MySQL with little or no degradation in performance.
Because you are asking this question, however, it makes me think that you may not have had a great deal of database experience.
I would suggest therefore that you get somebody with some good database experience to take a look at your database design to make sure that it will scale up effectively. You might also look into books and articles on database normalization. Normalizing your database design will definitely improve your chances for good performance.