A new method for crystal growth under reduced gravity conditions is proposed. The method combines the advantages of the Bridgman and floating zone techniques. The development of the method arose from the desire to obtain a melt configuration that is contained in a cylindrical ampoule but does not contact the lateral ampoule walls. Thus, allowing more control over the melt zone geometry. The free melt surface is attached to the growing crystal and is pinned to the edge formed by the end of the cylinder and the lateral walls. To successfully implement the technique it is necessary to define the size and shape of stable melt volumes that can be achieved in practice. This has been done under the assumption that the conditions that determine the shape of the free surface of a given melt volume in a typical temperature gradient are essentially identical to that of an isothermal capillary liquid. We have analyzed the shape and stability of the free surface of an isothermal melt volume in the above configuration and considered cases of zero-gravity and a steady acceleration that is parallel to the cylinder axis. Our treatment of the above problems has enabled us to define the range of system parameters for which stable melt configurations can be obtained and the method can be realized in practice.