This paper introduces a robust second-order low-rank BUG (Basis Update & Galerkin) integrator based on the midpoint rule for dynamical low-rank approximation of large matrix differential equations. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with an augmented basis. The method is designed to handle small singular values and large time derivatives of orthogonal factors in the low-rank matrix representation robustly. The authors prove a robust second-order error bound that shows improved dependence on the normal component of the vector field. The rigorous convergence analysis is supported by numerical experiments on the heat equation, a non-stiff discrete Schrödinger equation, and the Vlasov equation, demonstrating the effectiveness and accuracy of the proposed method. The paper also discusses variants of the integrator and compares it with other methods, such as projector-splitting integrators and explicit projected Runge-Kutta schemes.This paper introduces a robust second-order low-rank BUG (Basis Update & Galerkin) integrator based on the midpoint rule for dynamical low-rank approximation of large matrix differential equations. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with an augmented basis. The method is designed to handle small singular values and large time derivatives of orthogonal factors in the low-rank matrix representation robustly. The authors prove a robust second-order error bound that shows improved dependence on the normal component of the vector field. The rigorous convergence analysis is supported by numerical experiments on the heat equation, a non-stiff discrete Schrödinger equation, and the Vlasov equation, demonstrating the effectiveness and accuracy of the proposed method. The paper also discusses variants of the integrator and compares it with other methods, such as projector-splitting integrators and explicit projected Runge-Kutta schemes.