Wyner's Common Information and a natural relaxation are studied in the special case of Gaussian random variables. The relaxation replaces conditional independence by a bound on the conditional mutual information. The main contribution is the proof that Gaussian auxiliaries are optimal, leading to a closed-form formula. As a corollary, the proof technique also establishes the optimality of Gaussian auxiliaries for the Gaussian Gray-Wyner network, a long-standing open problem.