Binding energies and wave functions are calculated for the ground-state exciton in T-shaped quantum wires. It is shown that, if Coulomb interaction is taken into account, the hole is strongly localized by correlation with the electron. We demonstrate that no one-dimensional hole confinement is necessary for the formation of a one-dimensional exciton and that the exciton is roughly described by a two-dimensional hole, bound to a one-dimensional electron. Reasons are given for the shortcoming of the product Ansatz and the subband expansion for the pair wave function. For symmetric T structures with a thickness in the range from 5.4 to 7.0 nm, the calculated binding energies are larger than the values from approximate treatments, but smaller than the experimental results.