Let Σ = Σ_g,1 be a compact surface of genus g at least 3 with one boundary component, Γ its mapping class group and M = H₁(Σ, ℤ) the first integral homology of Σ. Using that Γ is generated by the Dehn twists in a collection of 2g+1 simple closed curves (Humphries' generators) and simple relations between these twists, we prove that H¹(Γ, M) is either trivial or isomorphic to ℤ. Using Wajnryb's presentation for Γ in terms of the Humphries generators we can show that it is not trivial.