Will a 7 speed cassette fit on a frame designed for 8 speed casette?

yup. You might have to adjust your derailleur. You may not get it to shift as perfectly, but you probably will. For a time in production there were several brands competing for the mountain bike market. the shift from 6 speed to 7 went smoothly, but when they went 8 there was campy, suntour, sram (then called grip shift) and of course shimano in the game. For a while a lot of derailleurs were sold as 7 or 8 speed derailleurs. The 8 speed coggs were made to fit on the old 7 speed cassette hubs:

SO... Yeah it should work. You will have to adjust your derailleur though.

The over locknut spacing of the 7 speed wheel might be too narrow for your frame, so you might have to add spacers for it to fit. 7 speed cassettes are over all a bit narrower than 8 speed cassettes, so you would have to adjust the limit screws on the derailleur. The big problem will be your shifters. 8 speed shifters don't shift the chain over as far each time you shift as 7 speed shifters, so with each shift your chain will be more and more out of alignment with the cog, until it won't shift at all. Shimano 7 speed cassettes use 5.0 mm center to center cog spacing, while 8 speed use 4.8 mm spacing. The answer that states that adding a single spacer is wrong. At best you might get 3 or 4 speeds to work......sort of. If you acquired this wheel to save money, there will be no savings, to make it work you will have to buy new shifters, and if your chain is old, a new chain as well. The derailleur would not be a problem

You will have to use a spacer on the 7 speed cassette but the shifter will work. If you are talking about a freewheel you would be down grading your bike. Cassettes need a free hub.

If you bought the complete new wheel set up, etc. - yes it will work. However, you will also need to change over from a 8 speed shifter to a 7 speed shifter.

It will likely fit, however you will need to change the chain, shifter and derailleur in order to make it work. If you JUST change the cassette, you will be in for serious trouble.