Dishwasher keeps running

The dishwasher should heat the water to about 50 C as the cycle calls for it. In these cycles, the dishwasher pauses after it has filled with water and waiting for the water to reach the temperature. In your case it look like the water isn't being heated (because of a problem with the heating element or thermostat), after the unit stops at the heating cycle, it never continues.
Another possible is that the timer unit is defective (this is the worse case because of its cost !!!!)

I have a similar problem. On the main program the dishwasher starts but stops during the cycle leaving the motor running. I have had the problem before and called a repair man out, he traced the fault to a broken wire leading from the control unit / timer to somewhere under the machine. The break was located where the wires went under the door, I would tend to suggest that this fault has re occurred and your fault may be the same.

DEFINATELY SEEMS LIKE THE HEATER. VERY SELDOM DOES ONE HAVE TO REPLACE THE TIMER. IF YOU FORWARD THE TIMER PAST THAT POINT OF NO MOVEMENT, DOES THE TIMER STILL NOT PROCEED OR DOES IT THEN CONTINUE. IF IT CONTINUES, MOST LIKELY HEATER FAULT IF NO MOVEMENT, INDICATES TIMER FAULT

90%of the time your element has not come on thus not allowing the timer to continue. remove the water, tilt the unit backwards and check underneath if there is an element reset button. if so depress it. try again. otherwise one needs to test the element for a resistance reading.(23 to 30ohms). no reading replace the element. try again.

Try adding some water with a bucket when the machine stops progressing. (about 1 or 2liters will do). Dont switch the machine off, just pause or open the door. Dovi is correct stating that the machine waits for the water to heat up. Sometimes there is not enough water in the machine and the heater-safty swith in the heater chaimber breaks the power to prevent element burnout.

Allways unplug the dishwasher when investigating faults. Next remove lid and right side panel, check the connections that go to the heating element. Some times the electrical connection makes a poor conection to the element, this causes heat to build on the connection tab. As time goes on, this effect gets worse, until finally is becomes open circuit from the element. Note that it is not the element that is faulty, only the piece of wire and spade connection that requires replacing. 50 pence not £5o !!!, so always check before calling an Engineer in. Hope this helps and saves you money.
Regards Ron the wizard

where is the thermostat located on the frigidaire 600 series? model #fdb635rbs0