As　a　common　generalization　of　bipartite　and　split　graphs,　monopolar　graphs　are　defined　in　terms　of　the　existence　of　certain　vertex　partitions.　It　has　been　shown　that　to　determine　whether　a　graph　has　such　a　partition　is　an　NP-complete　problem　for　general　graphs,　and　is　polynomial　time　solvable　for　several　classes　of　graphs.　In　this　paper,　we　investigate　graphs　that　admit　a　unique　such　partition　and　call　them　uniquely　monopolar-partitionable　graphs.　By　employing　a　tree　trimming　technique,　we　obtain　a　characterization　of　uniquely　monopolar-partitionable　block　graphs.　Our　characterization　implies　a　polynomial　time　algorithm　for　determining　whether　a　block　graph　is　uniquely　monopolar-partitionable.