A self-consistent approach to the polymer excluded-volume problem is proposed, based on the Domb-Gillis-Wilmers (DGW) probability distribution for all atom pairs W(r(j))(proportional to)rho(j)(theta)exp(-D rho(j)(delta)), rho(j)=r(j)/[r(j)(2)](1/2),r(j) being the interatomic distance between atoms separated by j bonds. As usual, we take A[r(j)(2)]proportional to j(2 nu) for j much greater than 1. The DGW distribution represents the simplest realistic assumption beyond the Gaussian distribution and is well supportedby both theoretical and numerical evidence. Upon optimization of the chain free energy with respect to the mean-square Fourier amplitudes, the numerical values theta=0.31 and nu=0.604 are obtained, in rough agreement with the expected results theta(c)ongruent to 0.7, nu=0.588. The results are shown to be invariant with the type of the polynomial expansion yielding the potential of mean force; the numerical disagreement may be due to the assumption of a single, average value of theta for all atomic pairs, consistent with a configurationally uniform chain model. Unlike with the Gaussian approximation, no factors comprising ln(j), ln[ln(j)]... are obtained within the expression of(r(j)(2)). Constraining to the values required by the DGW distribution the quadratic and quartic averages of the Fourier components constructed with the chain bond vectors, we obtain an approximate distribution function with a quartic polynomial of the chain coordinates within the exponent; this polynomial is proportional to the potential of mean force, useful in chain dynamics.