Bicycle Mechanics - New Wheelset, do I need a new cassette?

Can any one tell me if I can use my 6 speed cassette with these rims

http://www.cycleclubsports.com/p-27785-fsa-rd-60-700c-wheelset-2024-shimano-black.aspx

Or do I need to go the whole 9 yards and do a new cassette and new shifters? I have not purchased the wheels yet?

Those are not "rims", they are complete wheels. And the rear hub is going to fit a bike with 130 mm dropout spacing (modern road bike), and designed to accept a Shimano 8, 9, or 10-speed cassette. Since you have a 6-speed it's a pretty safe bet that the dropout spacing is less than 130. If your bike is steel it can be spread to accept these wheel. But if your shifters are indexed the rear shifter will not index with 8, 9, or 10-speeds. If your shifters are friction or can be switched to friction then the indexing is not a problem provided that the rear derailleur can swing wide enough to reach all cogs.
Notice that there are a lot of "ifs" in my statements because you have not provided enough information about your bike.
My advice to you would be "if" you need new wheels have wheels handbuilt for your bike. If your hubs are in good shape they can be used in the new wheels.
Are you sure your rear hub is a cassette/freehub and not a freewheel. Most 6-speeds are freewheels, not cassettes.

Are you sure your rear hub is a cassette/freehub and not a freewheel. Most 6-speeds are freewheels, not cassettes.

This would be my first concern.

+1 My guess is that you currently have a freewheel, and not a cassette. A lot of six speed bikes have freewheels. +1 Spacing will be wrong as well.