If the belt is still intact, there are two common conditions that can cause a 'no-spin' condition:
1. Lid Switch, this is a safety device that prevents injury if the lid is opened during the spin, the condition is as you described, there is no spin in the spin cycle and the motor is running.
2. A bad clutch, the clutch engages when the machine is in spin, if it is worn or damaged, it cannot engage and there will be no spin. Usually this condition is a problem where the spin is not satisfactory or slow and continues to get worse over time.
My best guess for you is a lid switch, which is easy to repair and it is inexpensive. I have written many hubs on appliance repair, you might check some of them out.