  We present an explicit evaluation of the double Gauss sum $\displaystyle
G(a,b,c;S;p^n):=\sum_{x,y=0}^{p^n-1} e^{2\pi i S(ax^2+bxy+cy^2)/p^n}$, where
$a, b, c$ are integers such that $\gcd(a,b,c)=1$, $p$ is a prime, $n$ is a
positive integer, and $S$ is an integer coprime to $p$.
