Using the shortest possible route (Highway 16 west to Jasper, Highway 93 [the Columbia Icefield Parkway] south to Highway 1 [the TransCanada Highway], Highway 1 east to Calgary, and Highway 2 (the Queen Elizabeth II] north to Edmonton) your total distance would be 915 kilometers (571 miles).
Driving time would vary, but Edmonton to Jasper is about 3.5 hours. Jasper to Calgary is about 5 hours. Calgary to Edmonton is about 2.8 hours.
So without stops for gas or meals-and assuming you don't hit Calgary during rush hour-our total driving time is 11.3 hours, more or less. I would think it's closer to 13 hours if you obey the speed limits (approximately) and stop to gas up.
Nathan
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It is 227 miles and an estimated driving time of 4 hours and 12 minutes according to Google Maps.