On bending of a convex surface to a convex surface with prescribed spherical image

We prove the following theorem. Let $F$ be a regular convex surface homeomorphic to the disk. Suppose the Gaussian curvature of $F$ is positive and the geodesic curvature of its boundary is positive as well. Let $G$ be a convex domain on the unit sphere bounded by a smooth curve and strictly contained in a hemisphere. Let $P$ be an arbitrary point on the boundary of $F$ and $P^*$ be an arbitrary point on the boundary of $G$. If the area of $G$ is equal to the integral curvature of the surface $F$, then there exists a continuous bending of the surface $F$ to a convex surface $F'$ such that the spherical image of $F'$ coincides with $G$ and $P^*$ is the image of the point in $F'$ corresponding to the point $P\in F$ under the isometry.