Behavioral robustness is essential in mechanism design. Existing papers focus on robustness as captured by dominant strategies. This paper studies the novel concept of externality-robustness, which addresses players' motives to affect other players' monetary payoffs. One example is externalities due to spite, which has been used to explain overbidding in second-price auctions. We show theoretically and experimentally that a trade-off exists between dominant-strategy implementation and externality-robust implementation. In particular, we derive the externality-robust counterpart of the second-price auction. Our experiments replicate the earlier finding of overbidding in the second-price auction, but we find that average bids equal value in the externality-robust auction. Our data also reveal that both auctions produce the same level of efficiency, suggesting that both dimensions of robustness are equally important. Our results are relevant for mechanism design in general, because the concept of externality-robustness is applicable to arbitrary mechanism design problems.