Primary cosmic-ray elemental spectra have been measured with the balloon-borne Cosmic Ray Energetics And Mass (CREAM) experiment since 2004. The third CREAM payload (CREAM-III) flew for 29 days during the 2007–2008 Antarctic season. Energies of incident particles above 1 TeV are measured with a calorimeter. Individual elements are clearly separated with a charge resolution of ~0.12 e (in charge units) and ~0.14 e for protons and helium nuclei, respectively, using two layers of silicon charge detectors. The measured proton and helium energy spectra at the top of the atmosphere are harder than other existing measurements at a few tens of GeV. The relative abundance of protons to helium nuclei is 9.53 ± 0.03 for the range of 1 TeV/n to 63 TeV/n. This ratio is considerably smaller than other measurements at a few tens of GeV/n. The spectra become softer above ~20 TeV. However, our statistical uncertainties are large at these energies and more data are needed.