The method of Padé truncation of perturbation expansions for thermodynamic potentials of molecular liquids is extended to the calculation of the solvation chemical potential of an infinitely dilute dipolar solute in a dipolar liquid. The Padé form is constructed to include nonlinear solvation effects of dipolar saturation at large and the linear response quadratic solute dipole dependence at small solute dipoles. The theory can accommodate polarizable solvents. The limiting case of electronically rigid solvent molecules is tested on the nonlinear reference hypernetted chain (RHNC) approximation for dipolar liquids. At high solvent polarities the Padé solvation chemical potential exceeds that of the RHNC. For both treatments, the nonlinear solvation contribution is found to pass through a maximum as a function of solvent polarity indicating that orientational saturation created by the solute breaks down with increasing solvent-solvent dipolar coupling. The Padé form of the chemical potential provides an analytical solution applicable to spectroscopic and electron transfer calculations involving solvation of fictitious complex-valued dipoles.