In a previous result concerning Meyniel graphs, the authors defined an operation for composing graphs. This operation constructs from two graphs G₁ and G₂ a new graph G₁G₂, called the amalgam of G₁ and G₂. If G₁ and G₂ are Meyniel graphs then G₁ G₂ is also a Meyniel graph. Similarly if G₁ and G₂ are perfect graphs then G₁G₂ is also a perfect graph. Here we study this operation from a polyhedral point of view. Starting from descriptions of the stable set polytopes P(G₁) and P(G₂) associated to G₁ and G₂ by means of valid linear inequalities, we give a complete description of P(G₁G₂). This result contains earlier results of this sort for the join, substitution composition, and clique join-operation. Furthermore it yields another proof that the amalgam operation preserves perfectness.