When basaltic magma is emplaced into continental crust, melting and generation of granitic magma can occur. We present experimental and theoretical investigations of the fluid dynamical and heat transfer processes at the roof and floor of a basaltic sill in which the wall rocks melt. At the floor, relatively low density crustal melt rises and mixes into the overlying magma, which would form hybrid andesitic magma. Below the roof the low-density melt forms a stable layer with negligible mixing between it and the underlying hotter, denser magma. Our calculations applied to basaltic sills in hot crust predict that sills from 10-1500 m thick require only 2-200 years to solidify, during which time large volumes of overlying layers of convecting silicic magma are formed. These time scales are very short compared with the lifetimes of large silicic magma systems of around 106 years, and also with the time scale of 107 years for thermal relaxation of the continental crust. An important feature of the process is that crystallisation and melting occur simultaneously, though in different spots of the source region. The granitic magmas formed are thus a mixture of igneous phenocrysts and lesser amounts of restite crystals. Several features of either plutonic or volcanic silicic systems can be explained without requiring large, high-level, long-lived magma chambers.