Yes, this follows from the analytic class number formula. See http://www.math.niu.edu/~rusin/known-math/97/sign .
Added I have now found a reference. This is a theorem of Mordell:
L. J. Mordell, The congruence $(p - 1/2)! \equiv \pm 1 (\operatorname{mod} p)$, American Mathematical Monthly, 68 (1961), 145-146. http://www.jstor.org/stable/2312481