We present a stable and convergent mixed finite element method (MFEM) for the linear regularized 13-moment (R13) equations in rarefied gas dynamics, originally derived by Struchtrup and Torrilhon. Unlike existing methods that often require stabilization via penalty terms to suppress spurious oscillations, our scheme achieves inherent stability by enriching the finite element basis with specific bubble functions. Based on previous works, we provide a rigorous theoretical analysis, establishing the discrete inf-sup condition and proving second-order convergence rates in the $L^2$ norm under mild regularity assumptions. To validate the theoretical results, we conduct numerical experiments on standard benchmarks, including Couette-Fourier flow and thermally-induced edge flow. Comparisons with standard schemes (e.g., Taylor-Hood elements) demonstrate that our proposed method yields robust numerical results even in the presence of geometric singularities and high rarefaction, overcoming the instability issues observed in traditional formulations.