We consider the problem of source coding subject to a fidelity criterion for the Gray-Wyner network that connects a single source with two receivers via a common channel and two private channels. The pareto-optimal trade-offs between the sum-rate of the private channels and the rate of the common channel is completely characterized for jointly Gaussian sources subject to the mean-squared error criterion, leveraging convex duality and an argument involving the factorization of convex envelopes. Specifically, it is attained by selecting the auxiliary random variable to be jointly Gaussian with the sources.