calculus - Computing the integral of $e^{-x^2}$ over the entire line

At lunch with a math friend years ago, he showed me an integral whose solution was, he said, so beautiful that it compelled him to become a professional mathematician. I scribbled the integral down on a napkin, but for the life of me I cannot remember the trick he found so compelling.

And these things are hard to Google...

$$ \int_{-\infty}^{\infty} e^{-x^2} dx $$