Search auctions have become a dominant source of revenue generation on the Internet. Such auctions have typically used per-click bidding and pricing. We propose the use of hybrid auctions where an advertiser can make a per-impression as well as a per-click bid, and the auctioneer then chooses one of the two as the pricing mechanism. We assume that the advertiser and the auctioneer both have separate beliefs (called priors) on the click-probability of an advertisement. We first prove that the hybrid auction is truthful, assuming that the advertisers are risk-neutral. We then show that this auction is superior to the existing per-click auction in multiple ways: 1) It takes into account the risk characteristics of the advertisers. 2) For obscure keywords, the auctioneer is unlikely to have a very sharp prior on the click-probabilities. In such situations, the hybrid auction can result in significantly higher revenue. 3) An advertiser who believes that its click-probability is much higher than the auctioneer's estimate can use per-impression bids to correct the auctioneer's prior without incurring any extra cost. 4) The hybrid auction can allow the advertiser and auctioneer to implement complex dynamic programming strategies. As Internet commerce matures, we need more sophisticated pricing models to exploit all the information held by each of the participants. We believe that hybrid auctions could be an important step in this direction.