This　paper　considers　the　problem　of　minimizing　the　sum　of　a　smooth　convex　function　and　a　separable　convex　function　subject　to　linear　coupling　constraints.　Problems　of　this　form　arise　in　many　contemporary　applications　including　signal　processing,　wireless　networking　and　smart　grid　provisioning.　We　propose　a　new　algorithm　called　the　proximal　block　minimization　method　of　multipliers　with　a　substitution　to　solve　this　family　of　problems.　The　proposed　algorithm　integrates　the　block　coordinate　descent　and　alternating　direction　method　of　multipliers　with　substitution.　For　the　proposed　algorithm,　we　prove　its　convergence　via　the　analytic　framework　of　contractive　type　methods　and　we　derive　a　worst-case　Omicron(1/t)　convergence　rate　in　an　ergodic　sense,　where　t　denotes　the　number　of　iterations.　Finally,　some　preliminary　numerical　results　are　reported　to　support　the　efficiency　of　the　new　algorithm.