An $\mathcal{O}(N^2)$ algorithm for the convolution of compactly supported Legendre series is described. The algorithm is derived from the convolution theorem for Legendre polynomials and the recurrence relation satisfied by spherical Bessel functions. Combining with previous work yields an $\mathcal{O}(N^2)$ algorithm for the convolution of Chebyshev series. Numerical results are presented to demonstrate the improved efficiency over the existing algorithm.