Differential cross sections have been measured near 0{degree} for the dominant channels in the mirror reaction {sup 13}C({sup 13}N,{sup 13}C){sup 13}N at {ital E}/{ital A}=57 and 105 MeV. The cross sections of the peaks in the excitation spectrum are discussed in terms of Gamow-Teller and Fermi transition strength. The cross section per unit Gamow-Teller strength is found to be enhanced relative to that for unit Fermi strength when compared with previous results from ({ital p},{ital n}) reactions. The present work represents the first use of a mirror-symmetric projectile and target system in a charge-exchange reaction as well as the first application of the developing radioactive nuclear beam field to this area. {copyright} {ital 1995 The American Physical Society.}