Double Integrals · Superchallenge problem

Evaluate $\displaystyle\int_0^{1}\!\int_0^{1}\dfrac{dx\,dy}{\sqrt{1 - x^2y^2}}$. You may use $\displaystyle\int_0^{\pi/2}\ln(\sin t)\,dt = -\dfrac{\pi}{2}\ln 2$. Give an exact answer.