Let \(\mathcal{F}(X)\) be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that \(\Gamma\) is a closed rectifiable Jordan curve in \(\mathbb{R}^{3}\). We then prove that there is a conformally parametrized minimizer of \(\mathcal{F}\) in the class \(\mathcal{C}(\Gamma)\) of surfaces \(X \in H^{1,2}(B, \mathbb{R}^{3}) \cap C^{0}(\overline{B}, \mathbb{R}^{3})\) of the type of the disk B which are bounded by \(\Gamma\). An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on \(\epsilon\)-conformal mappings. In addition we show that the minimizer of \(\mathcal{F}\) is Holder continuous in B, and even in \(\overline{B}\) if \(\Gamma\) satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one.