Wheel sizes generally aren't interchangable for a particular frame. A 700c wheel is somewhat larger than a 650c.
The only place I've seen a wheel size used on a bike not originally designed for it, was a custom-made 650c wheelset built for a mountain bike (which normally takes 26" tyres), so it could use skinny road slicks. This bike had disc brakes, so brake caliper reach etc. was a non-issue.
Depending on how old the bike is, it may have a different sized rim (i.e. neither 700c nor 650c.) Are there any markings on the tyre that's on it right now?