We present an efficient method for seeking the numerical solution of a Caputo- type diffusion equation with a variable coefficient. Since the solution of such an equation is likely to have a weak singularity near the initial time, the time-fractional derivative is discretized using the L1 formula on nonuniform meshes. For spatial derivative, we employ the local discontinuous Galerkin method to derive a fully discrete scheme. Based on a dis- crete fractional Gronwall inequality, the numerical stability and convergence of the derived scheme are proven which are both α-robust, that is, the bounds obtained do not blow up as α → 1⁻. Finally, numerical experiments are displayed to confirm the theoretical results.