We consider measure-preserving diffeomorphisms of the two-dimensional torus with zero entropy. We prove that every ergodic C^3-diffeomorphism f of the two-dimensional torus with linear growth of the derivative (i.e. the sequence \{n^{-1}Df^n\}_{n\in{\mathbb N}} is uniformly separated from 0 and \infty and it is bounded in the C^2-norm) is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle C^3-cocycle with non-zero topological degree.