No, you don't.
Most statistics master's program (biostatistics included) will admit people with non-mathematics master's who have the appropriate prerequisites. At minimum, this will include the calculus sequence (calculus I-III) and linear algebra. Commonly, they also require one or two calculus-based statistics and probability courses. The more math classes you have (and computer science - a programming language is a huge plus, since many biostatistician jobs want people who can program) the better your chances. Some biostatistics programs will conditionally admit you without a few of the prerequisites and allow you to take them before you start the program classes, but I think this is generally done for people with science backgrounds and strong credentials who do not have one or two math requirements.
When polarscribe says contact schools, he means look at the websites. The vast majority of statistics programs have their prerequisites explicitly listed on the pages. But EVERY statistics program will require, at minimum, the calculus sequence and linear algebra.