Convolution of two Gaussians is a Gaussian

1. the Fourier transform (FT) of a Gaussian is also a Gaussian
2. The convolution in frequency domain (FT domain) transforms into a simple product
3. then taking the FT of 2 Gaussians individually, then making the product you get a (scaled) Gaussian and finally taking the inverse FT you get the Gaussian

Fourier Transform will help you out to conclude that the convolution is also a gaussian.

See this for two common alternatives.