The high-frequency O-H stretching vibrational overtone spectrum of the water trimer has been studied computationally. The water trimer has been approximated as three individually vibrating water monomers. An internal coordinate Hamiltonian is constructed for each triatomic monomer unit using an exact kinetic-energy operator within the Born-Oppenheimer approximation and the two O-H distances and the H-O-H bending angle as the internal coordinates. The potential-energy surface, PES, is calculated using the coupled cluster method with single, double, and perturbative triple excitations, CCSD(T), and augmented correlation-consistent polarized valence triple-zeta, aug-cc-pVTZ, basis set. The PES is improved applying the counterpoise correction. The finite differences method is used to obtain the dipole moment surface. Eigenvalues of the Hamiltonians are computed with the variational method. Finally, matrix isolation infrared data are used to optimize some PES parameters to improve the results.