The basis-set convergence of the correlation contribution to the two-electron Darwin term with the cardinal number(X) of the basis set in conventional calculations is investigated for He, H-2, Ne, and HF. For He, the form D-corr(2e)(X) = D-0(2e) + C1X-1 describes the convergence for numerically optimized basis sets of the principal-expansion type accurately, and, for the larger of these basis sets, two-point extrapolated values obtained with this X-1 form are within 0.1% of the basis-set limit established with the Hylleraas expansion. For all four systems, the X-1 form describes the convergence of the correlation-consistent cc-pVXZ basis sets accurately. Accordingly, for these basis sets, two-point X-1-extrapolated values represent significant improvements on the unextrapolated results.