From Uniform to 2D gaussian

I have a 2D uniform distribution. I would like to assign probability-weights in the distribution, according to the standard deviation to convert the distribution into a 2D gaussian.

I know how to do it in the 1D case, by splitting the points from -1 to 1 sigma, etc and assigning the probability as a weight but I haven't found a way to combine the probabilities from the 2 planes (eg at 1 sigma in x and 6 sigma in y)

If $G(x)$ is a 1D Gaussian, then $G(x) G(y)$ is a 2D Gaussian.