We report the abundances of neon isotopes in the Galactic cosmic rays (GCRs) using data from the Cosmic Ray Isotope Spectrometer (CRIS) aboard the Advanced Composition Explorer (ACE). These abundances have been measured for seven energy intervals over the energy range of 84 ≤ E/M ≤ 273 MeV nucleon^-1. We have derived the ²²Ne/²⁰Ne ratio at the cosmic-ray source using the measured ²¹Ne, ¹⁹F, and ¹⁷O abundances as "tracers" of secondary production of the neon isotopes. Using this approach, the ²²Ne/²⁰Ne abundance ratio that we obtain for the cosmic-ray source is 0.387 ± 0.007(statistical) ± 0.022(systematic). This corresponds to an enhancement by a factor of 5.3 ± 0.3 over the ²²Ne/²⁰Ne ratio in the solar wind. This cosmic-ray source ²²Ne/²⁰Ne ratio is also significantly larger than that found in anomalous cosmic rays, solar energetic particles, most meteoritic samples of matter, and interplanetary dust particles. We compare our ACE CRIS data for neon and refractory isotope ratios, and data from other experiments, with recent results from two-component Wolf-Rayet (W-R) models. The three largest deviations of GCR isotope ratios from solar system ratios predicted by these models, ¹²C/¹⁶O, ²²Ne/²⁰Ne, and ⁵⁸Fe/⁵⁶Fe, are indeed present in the GCRs. In fact, all of the isotope ratios that we have measured are consistent with a GCR source consisting of about 80% material with solar system composition and about 20% W-R material. Since W-R stars are evolutionary products of OB stars, and most OB stars exist in OB associations that form superbubbles, the good agreement of these data with W-R models suggests that superbubbles are the likely source of at least a substantial fraction of GCRs.