Consider the random graph sampled uniformly from the set of all simple graphs with a given degree sequence. Under mild conditions on the degrees, we establish a Large Deviation Principle (LDP) for these random graphs, viewed as elements of the graphon space. As a corollary of our result, we obtain LDPs for functionals continuous with respect to the cut metric, and obtain an asymptotic enumeration formula for graphs with given degrees, subject to an additional constraint on the value of a continuous functional. Our assumptions on the degrees are identical to those of Chatterjee, Diaconis and Sly (2011), who derived the almost sure graphon limit for these random graphs.