I recently discovered that Star Trek the next generation is on Netflix, of course that means watching the whole series from the begining.

In episode 5 of season 1, Jordi makes a comment which can be used to determine not only the maximum speed of the Enterprise but how fast a warp is.
Jordi says "We've traveled 2,750,000 light years Captain. At max speed [warp 10] it will take us 300 years to get back!".

So according to Jordi at max warp the enterprise can travel 2.75 million light years in 300 years. That means the Enterprise can travel 9,166.66 light years in 1 year (2,750,000/300).

Which calculates to:
25.11 light years a day (9,166.66/365)
1.04 light years an hour (25.11/24)

A light year is equal to 670,616,629mph.


So at max warp the Enterprise can travel at 701,748,755.636mph (670,616,629*1.04).


Since max warp is warp 10, a single warp is 70,174,487.55 mph. (701,748,755.636/10)

EDIT: It seems that the writers of that episode did not do thier homework, as warp 1 is equal to the speed of light. Althought fun to figure out, my calculatsions are therefore incorrect. Checkout this article for more information http://en.memory-alpha.org/wiki/Warp_factor#Star_Trek:_The_Next_Generation