The stabilization of one-dimensional solitons by a nonlinear lattice against the critical collapse in the focusing quintic medium is a challenging issue. We demonstrate that this purpose can be achieved by combining a nonlinear lattice and saturation of the quintic nonlinearity. The system supports three species of solitons, namely, fundamental (even-parity) ones and dipole (odd-parity) modes of on- and off-site-centered types. Very narrow fundamental solitons are found in an approximate analytical form, and systematic results for very broad unstable and moderately broad partly stable solitons, including their existence and stability areas, are produced by means of numerical methods. Stability regions of the solitons are identified by means of systematic simulations. The stability of all the soliton species obeys the Vakhitov-Kolokolov criterion.