We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let F : B n → C n be a univalent map from the unit ball whose image D is convex. Let S ⊂ ∂B n be the set of points ξ such that lim z→ξ F (z) = ∞. Then we prove that S is either empty, or contains one or two points and F extends as a homeo-morphism˜Fmorphism˜ morphism˜F : B n \ S → D. Moreover, S = ∅ if D is bounded, S has one point if D has one connected component at ∞ and S has two points if D has two connected components at ∞ and, up to composition with an automorphism of the ball and renormalization, F is an extension of the strip map in the plane to higher dimension.