The author proves Lipschitz continuity of local minimizers \(u: \mathbb{R}^n\to \mathbb{R}\) of the variational integral \(\int_\Omega f(\nabla u) dx\) where the integrand is assumed to be an even convex function from \(\mathbb{R}^n\to [0,\infty)\) satisfying \(f(\xi)\geq c_0|\xi|^2\), \(\lim_{\xi\to 0} {1\over|\xi|} f(\xi)= 0\) and \(Df(\xi)\cdot \eta\leq p{f(\xi)\over |\xi|} |\eta|\) with numbers \(c_0> 0\) and \(p\geq 2\). A model case is given by \(f(\xi)= |\xi|^2+ {|\xi|^p\over 1+|\xi|} \sum^n_{i=1} |\xi_i|\). The main ingredient of the proof is an approximation argument which is used to overcome the lack of smoothness of the integrand. It should be noted that no upper bound on \(p\) is needed.