A time-stepping scheme for subdiffusion equation with a Riemann--Liouville time fractional derivative is developed and analyzed. This is the first paper to show that the scheme for the model problem under consideration is second-order accurate (sharp error estimate) over nonuniform time steps. The established convergence analysis is novel and concise. For completeness, the scheme is combined with the standard Galerkin finite elements for the spatial discretization, which will then define a fully discrete numerical scheme. The error analysis for this scheme is also investigated. To support our theoretical contributions, some numerical tests are provided at the end. The considered (typical) numerical example suggests that the imposed time-graded meshes assumption can be further relaxed.