We have predicted that {ital B}({ital D}{sup *+}{r arrow}{ital D}{sup +}{gamma}) is between 0.4% and 2% using data from the similar decays of {ital D}{sup *0} and {ital K}{sup *} mesons. Our calculation is based on an effective field theory with expansions in the external momentum and heavy-quark mass. We also have calculated the {ital D}{sup *0} branching fractions and found reasonable agreement between theory and experiment. Our calculation suggests that nonleading terms in the quark-mass expansion may still be quite important for both {ital s} and {ital c} quarks.