We introduce a new meta-analysis estimator, the weighted and iterated least squares (WILS), that greatly reduces publication selection bias (PSB) when selective reporting for statistical significance (SSS) is present. WILS is the simple weighted average that has smaller bias and rates of false positives than conventional meta-analysis estimators, the unrestricted weighted least squares (UWLS), and the weighted average of the adequately powered (WAAP) when there is SSS. As a simple weighted average, it is not vulnerable to violations in publication bias corrections models' assumptions too often seen in application. WILS is based on the novel idea of allowing excess statistical significance (ESS), which is a necessary condition of SSS, to identify when and how to reduce PSB. We show in comparisons with large-scale preregistered replications and in evidence-based simulations that the remaining bias is small. The routine application of WILS in the place of random effects would do much to reduce conventional meta-analysis's notable biases and high rates of false positives. (PsycInfo Database Record (c) 2024 APA, all rights reserved).