  Given a closed oriented 3-manifold M, we establish an isomorphism between the
Heegaard Floer homology group HF^+(-M) and the embedded contact homology group
ECH(M). Starting from an open book decomposition (S,h) of M, we construct a
chain map \Phi^+ from a Heegaard Floer chain complex associated to (S,h) to an
embedded contact homology chain complex for a contact form supported by (S,h).
The chain map \Phi^+ commutes up to homotopy with the U-maps defined on both
sides and reduces to the quasi-isomorphism \Phi from "The equivalence of
Heegaard Floer homology and embedded contact homology I, II" on subcomplexes
defining the hat versions. Algebraic considerations then imply that the map
\Phi^+ is a quasi-isomorphism.
