Yielding in pure BCC metals and dilute substitutional alloys occurs by double-kink nucleation and propagation along screw dislocations. At low temperatures, the yield stress is controlled by double-kink nucleation. Here, an analytical statistical model is presented to predict the stress- and length-dependent double-kink nucleation barrier in dilute BCC alloys solely in terms of the double-kink process in the pure metal and the solute/screw-dislocation interaction energies in the dilute alloy. Consistent with early literature, dilute alloying always reduces the double-kink nucleation barrier (softening) independent of solutes or matrix. The model is extensively validated via simulations in model Fe-Si alloys described by interatomic potentials. The model is then compared to experiments on real Fe-Si, W-Ta, and W-Re alloys, showing qualitative agreement consistent with the accuracy of the inputs. A cross-over from the dilute limit to the non-dilute limit, where there is hardening, is analyzed using the present theory and the nondilute theory of Maresca et al. The analysis for Fe-Si is consistent with a cross-over at approximate to 2 - 3 at.%Si, as observed experimentally, and qualitatively consistent with W-Ta and W-Re. The present theory plus the recent theory of Maresca et al. together provide a coherent predictive framework for strengthening of screw dislocations over the full range of concentrations from extremely dilute (<< 1 at.%), to dilute (up to a few at.%) and non-dilute alloys including High Entropy Alloys. (C) 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.