The hydration enthalpy and Gibbs free energy of proton and hydroxide are calculated by means of a combination of ab initio density functional theory and a polarizable continuum model within the self-consistent reaction field method. The ion-water cluster models here used include up to 13 water molecules solvating the ions. This allows the first and second solvation shells to be described explicitly from first principles. Vibrational contributions to the enthalpy and entropy have been taken into account. Our best model of the hydrated proton includes three molecules in the first hydration shell and nine molecules in the second shell. The calculated proton hydration enthalpy is approximate to -1150 kJ/mol, which is in rather good agreement with the most recent results from cluster-ion solvation data. The hydration free energy of the proton has a larger error of approximate to 50-80 kJ/mol as compared to recently reported values. The calculated hydroxide hydration enthalpy, approximate to -520 kJ/mol, and hydration free energy, approximate to -400 kJ/mol, are consistent with well-established values taken from experiment. Two different sources of error in our calculations, namely, the nature of the hydrated complex and the outlying charge correction, are discussed. Moreover, we compare the results from three slightly different methods for the calculation of hydration energies.