Using recently calculated rate coefficients for reactive coupling of chains at polymer-polymer interfaces, we describe the diffusion-controlled growth of diblock copolymer at the interface between dissimilar molten homopolymer films. The growth exhibits several regimes, successively controlled by (1) the quasilocal reaction rate near the interface, (2) center-of-mass diffusion of reactive homopolymer to the interface, and (3) the diffusion barrier presented by the growing copolymer layer. Our results can be tested by time-dependent measurements of the copolymer coverage or concentration of reacted coupling groups.