The emergence of deep learning has stimulated a new class of partial differential equation (PDE) solvers in which the unknown solution is represented by a neural network. Within this framework, residual minimization in dual norms—central to weak adversarial neural network (WAN) formulations—leads naturally to saddle-point problems whose numerical behaviour depends critically on the iterative strategy used to resolve them. Recent studies have shown that the direct approaches for such formulations may suffer from algorithmic instability, motivating the development of more robust solution strategies grounded in classical numerical analysis. In this work, we propose the Uzawa Double Deep Ritz Method, a deep PDE solver that integrates variational neural approximations with the classical Uzawa iteration. The method is built around two coupled update rules performed at each iteration: a residual update, obtained by minimizing a Ritz functional associated with the dual problem, and a solution update, obtained by minimizing a Ritz functional driven by the current residual. Both variables are represented by neural networks, and the resulting scheme mirrors the structure of classical Uzawa methods for saddle-point problems while remaining fully mesh-free. The main contributions of this work are the formulation of the Uzawa Double Deep Ritz algorithm and a continuous-level convergence analysis. We establish convergence for an inexact Uzawa scheme in which both update steps are performed approximately, and prove that the iteration remains stable and convergent provided each update proceeds in a correct descent direction, even when the Ritz problems are not solved exactly. Numerical experiments are presented to illustrate and support the theoretical results and demonstrate the robustness and effectiveness of the proposed approach.