We investigate the Block GMRES method for solving large and sparse linear systems with multiple right-hand sides. For solving linear systems with a single right-hand side, the Weighted GMRES method based on the weighted minimal residual condition has been proposed as an improvement of the GMRES method. In this paper, by applying the idea of the Weighted GMRES method to the Block GMRES method, we propose a Weighted Block GMRES method. The numerical experiments indicate that the Weighted Block GMRES($m$) method has higher performance for efficient convergence than the Block GMRES($m$) method.