We present a new algorithm for computing the convolution of two compactly supported functions. The algorithm approximates the functions to be convolved using Fourier extensions and then uses the fast Fourier transform to efficiently compute Fourier extension approximations to the pieces of the result. The complexity of the algorithm is $\mathcal{O}\bigl(N (\log N)^2\bigr)$, where $N$ is the number of degrees of freedom used in each of the Fourier extensions.