A new implicit solvent model for Monte Carlo simulations of peptides and proteins is presented. The important features of the model are: electrostatics of polar solvation described by nonlinear, distance-dependent screened Coulomb potentials that are formally connected to classical dielectric theory of polar solvation; the self-energy is derived from the integral form of the Born equation; and the introduction of the Born radius of an atom embedded in a protein modeled as a dielectric continuum. The model was parametrized in the context of the CHARMM PAR22 force field by fitting to experimental solvation free energies of amino acid side chain analogues. Introduction of an intrinsic vacuum screening constant for each amino acid was required to obtain the correct screening behavior in solution. Solvation energies of several peptides in different conformations were calculated and compared with the corresponding quantities calculated with the Poisson-Boltzmann equation. The comparative results were highly correlated in all cases, and the correlation was independent of peptide length and charge, Moreover, the self-energy terms and the electrostatic interaction energies, separately calculated from the two methods, were also highly correlated. The computing time required for the model is about 5 times the vacuum calculation for small systems (similar to 100 atoms), but decreases to a factor of 3 as the system size increases. The development of a new hydrogen-bonding algorithm as well as initial results of using the new model for sequence-to-structure calculations on peptides is presented in the accompanying paper. The combination results in a complete continuum solvation model.