ABSTRACT. Let X ′ be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C + -action on it such that S = X ′ //C + is smooth. We prove that X ′ is isomorphic to S × C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C + -action on C 3 is a translation in a suitable coordinate system. this problem we consider a more general situation when there is a nontrivial algebraic C + -action on a complex three-dimensional affine algebraic variety X ′ such that its ring of regular functions is factorial and H 3 (X ′ ) = 0. By a theorem of Zariski [Za] the algebraic quotient X ′ //C + is isomorphic to an affine surface S. Let π : X ′ → S be the natural projection. Then there is a curve Γ ⊂ S such that for E = π −1 (Γ) the varietyThe study of morphism π| E : E → Γ is central for this paper. As an easy consequence of the Stein factorization one can show that π| E = θ • ϑ where ϑ : E → Z is a morphism into a curve Z with general fibers isomorphic to C, and θ : Z → Γ is a quasi-finite morphism. A more delicate fact (Proposition 3.2) is that θ is, actually, finite and, furthermore, if in addition X ′ is smooth and H 2 (X ′ ) = 0, then each irreducible component Z 1 of Z, such that θ| Z 1 is 1 The author was partially supported by the NSA grant MDA904-00-1-0016. Theorem 1. Every free algebraic C + -action on C 3 is a translation in a suitable polynomial coordinate system.