We introduce and analyze a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kind with or without known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind
problems, where the former builds on connections with Toeplitz operators.