Phonon sideband optical spectroscopy determines the energy of the dark K-momentum exciton for (6,5) carbon nanotubes. One-phonon sidebands appear in absorption and emission, split by two zone-boundary (K-point) phonons. Their average energy locates the E11 K-momentum exciton 36 meV above the E11 bright level, higher than available theoretical estimates. A model for exciton-phonon coupling shows the absorbance sideband depends sensitively on the K-momentum exciton effective mass and has minimal contributions from zone-center phonons, which dominate the Raman spectra of carbon nanotubes.