arXiv:2504.13273v3 Announce Type: replace-cross Abstract: Propensity scores near zero or one can make standard confidence intervals undercover. Econometric theory has targeted the Average Treatment Effect (ATE) under such weak overlap using nonstandard estimators or intervals. Practice has instead favored fixed thresholding rules that restore asymptotic normality by discarding extreme-propensity units for whom treatment may matter most. I show that standard t-tests for the ATE remain valid even when overlap is so weak that every regular estimator has infinite asymptotic variance. The key is to threshold the usual doubly robust estimator aggressively enough to control bias among unthresholded observations, but mildly enough to control bias from thresholding. I provide sufficient conditions for such thresholds to exist, and propose an adaptive rule of thumb to achieve coverage without knowing the weak overlap degree. In a canonical application, fixed trimming excludes 18% of the sample; the adaptive procedure estimates the resulting bias at five percent of the effect.