A metric $ds^2$ admits a $\Sigma$-realization if there is a realization of it in $E^3$ in the form of a surface whose boundary lies on a given surface $\Sigma$. This paper proves the existence of $\Sigma$-realizations of a certain class of metrics of positive curvature for surfaces of quite general form, and describes a number of possible $\Sigma$-realizations of the given metric. The proof is based on a consideration of a nonlinear boundary-value problem for immersion equations. Bibliography: 3 titles.