Depends on where EXACTLY you build it. When it starts to rain in the grassy areas it will snow in the frozen ones. However, there is usually an area of snow near the grass that still gets rain. If you build there or remove the snow from the ground it will never come back or get snow covered. If you move deeper into the snowfield eventually you will get to an area where it's snowing instead of raining. Any block with only air directly above it will eventually become snow covered.
I see them! THEIR over THEY'RE at THERE computers! The grammar police! And I've just enraged them!