An auction is externality-robust if unilateral deviations from equilibrium leave the other bidders' payoffs unaffected. The equilibrium and its outcome will then persist if certain types of externalities arise between bidders. One example are externalities due to spiteful preferences, which have been used to explain overbidding in the second-price auction (SPA). Another example are cross-shareholdings between companies that compete in an auction. We derive an auction that coincides with the SPA in terms of efficiency and revenue but, in contrast to the SPA, is externality-robust. The externality-robust auction (ERA) is a first-price auction in which truthful bidding is encouraged by bonus payments. We test the robustness property experimentally by comparing SPA and ERA. We replicate the earlier finding of significant average overbidding in the SPA, but we find that bidders bid on average their value in the ERA. We conduct additional treatments where bidders play against the computer and we use controls for cognitive skills and joy of winning to further pin down the reasons behind the subjects' bidding behavior.