Abstract: The method described previously for the solution of the wave equation of two-electron atoms has been applied to the $1^{1}S$ and $2^{3}S$ states of helium, with the purpose of attaining an accuracy of 0.001 ${\mathrm{cm}}^{\ensuremath{-}1}$ in the nonrelativistic energy values. For the $1^{1}S$ state we have extended our previous calculations by solving determinants of orders 252, 444, 715, and 1078, the last yielding an energy value of -2.903724375 atomic units, with an estimated error of the order of 1 in the last figure. Applying the mass-polarization and relativistic corrections derived from the new wave functions, we obtain a value for the ionization energy of 198 312.0258 ${\mathrm{cm}}^{\ensuremath{-}1}$, as against the value of 198 312.011 ${\mathrm{cm}}^{\ensuremath{-}1}$ derived previously from the solution of a determinant of order 210. With a Lamb shift correction of -1.339, due to Kabir, Salpeter, and Sucher, this leads to a theoretical value for the ionization energy of 198 310.687 ${\mathrm{cm}}^{\ensuremath{-}1}$, compared with Herzberg's experimental value of 198 ${310.8}_{2}$\ifmmode\pm\else\textpm\fi{}0.15 ${\mathrm{cm}}^{\ensuremath{-}1}$.For the $2^{3}S$ state we have solved determinants of orders 125, 252, 444, and 715, the last giving an energy value of -2.17522937822 a.u., with an estimated error of the order of 1 in the last figure. This corresponds to a nonrelativistic ionization energy of 38 453.1292 ${\mathrm{cm}}^{\ensuremath{-}1}$. The mass-polarization and relativistic corrections bring it up to 38 454.8273 ${\mathrm{cm}}^{\ensuremath{-}1}$. Using the value of 74.9 ry obtained by Dalgarno and Kingston for the Lamb-shift excitation energy ${K}_{0}$, we get a Lamb-shift correction to the ionization energy of the $2^{3}S$ state of -0.16 ${\mathrm{cm}}^{\ensuremath{-}1}$. The resulting theoretical value of 38 454.66 ${\mathrm{cm}}^{\ensuremath{-}1}$ for the ionization potential is to be compared with the experimental value, which Herzberg estimates to be 38 454.73\ifmmode\pm\else\textpm\fi{}0.05 ${\mathrm{cm}}^{\ensuremath{-}1}$. The electron density at the nucleus $D(0)$ comes out 33.18416, as against a value of 33.18388\ifmmode\pm\else\textpm\fi{}0.00023 which Novick and Commins deduced from the hyperfine splitting. We have also determined expectation values of several positive and negative powers of the three mutual distances, which enter in the expressions for the polarizability and for various sum rules.