There are many settings where researchers are interested in estimating average treatment effects and are willing to rely on the unconfoundedness assumption, which requires that the treatment assignment be as good as random conditional on pre-treatment variables. The unconfoundedness assumption is often more plausible if a large number of pre-treatment variables are included in the analysis, but this can worsen the performance of standard approaches to treatment effect estimation. In this paper, we develop a method for de-biasing penalized regression adjustments to allow sparse regression methods like the lasso to be used for sqrt{n}-consistent inference of average treatment effects in high-dimensional linear models. Given linearity, we do not need to assume that the treatment propensities are estimable, or that the average treatment effect is a sparse contrast of the outcome model parameters. Rather, in addition standard assumptions used to make lasso regression on the outcome model consistent under 1-norm error, we only require overlap, i.e., that the propensity score be uniformly bounded away from 0 and 1. Procedurally, our method combines balancing weights with a regularized regression adjustment.