This paper introduces a second-order robust basis-update & Galerkin (BUG) integrator for dynamical low-rank approximation based on the midpoint rule. The method is designed to handle the challenges of small singular values and large time derivatives in low-rank matrix representations. The integrator first performs a half-step using a first-order BUG integrator, followed by a Galerkin update with an augmented basis. The method is proven to have a robust second-order error bound, showing improved dependence on the normal component of the vector field. The paper discusses the application of the method to various problems, including the heat equation, a non-stiff discrete Schrödinger equation, and the Vlasov equation. Numerical experiments demonstrate the effectiveness of the midpoint BUG integrator, showing second-order convergence in time and better accuracy compared to first-order methods. The method is also compared with other integrators, such as the projected Runge-Kutta methods and the augmented BUG integrator, showing that the midpoint BUG integrator achieves robust second-order accuracy. The paper also provides a rigorous error analysis, showing that the method's error bound is independent of the singular values of the solution and its derivatives. The results are validated through numerical experiments, which confirm the robustness and efficiency of the midpoint BUG integrator in handling stiff and non-stiff problems. The method is expected to be extended to tree tensor networks in future work.This paper introduces a second-order robust basis-update & Galerkin (BUG) integrator for dynamical low-rank approximation based on the midpoint rule. The method is designed to handle the challenges of small singular values and large time derivatives in low-rank matrix representations. The integrator first performs a half-step using a first-order BUG integrator, followed by a Galerkin update with an augmented basis. The method is proven to have a robust second-order error bound, showing improved dependence on the normal component of the vector field. The paper discusses the application of the method to various problems, including the heat equation, a non-stiff discrete Schrödinger equation, and the Vlasov equation. Numerical experiments demonstrate the effectiveness of the midpoint BUG integrator, showing second-order convergence in time and better accuracy compared to first-order methods. The method is also compared with other integrators, such as the projected Runge-Kutta methods and the augmented BUG integrator, showing that the midpoint BUG integrator achieves robust second-order accuracy. The paper also provides a rigorous error analysis, showing that the method's error bound is independent of the singular values of the solution and its derivatives. The results are validated through numerical experiments, which confirm the robustness and efficiency of the midpoint BUG integrator in handling stiff and non-stiff problems. The method is expected to be extended to tree tensor networks in future work.