Adjustable Robust Optimization (ARO) is a paradigm for facing uncertainty in a decision problem, in case some recourse actions are allowed after the actual value of all input parameters is revealed. While several approaches have been introduced for the linear case, little is known regarding exact methods for the convex case. In this work, we introduce a new general method for solving ARO problems involving convex functions in the recourse problem. We first recall a semi-infinite reformulation of the problem and we show how to solve it by row generation, provided that one can tackle a non-convex separation problem. For the relevant case where the uncertainty set has an affine mapping to a 0-1 set, we show that the separation problem can be reformulated as a convex MINLP, thus allowing us to derive a computationally sound exact method. We then study the convergence of the obtained algorithm and apply it to a resource planning problem.