The ground state and first few excited energy levels of the generalized anharmonic oscillator defined by the HamiltonianH=−d2/dx2+x2+λx2k (k=3, 4,...) have been calculated by employing the method of quantum normal form, which is the quantum mechanical analogue of the classical Birkhoff-Gustavson normal form. The present energy eigenvalues are consistent with other tabulations of the energy levels.