Election audit procedures usually rely on precinctbased recounts, in which workers manually review all paper ballots from selected polling places, but these recounts can be expensive due to the labor required. This paper proposes an alternative audit strategy that allows machines to perform most of the work. Precincts are recounted using recounting machines, and their output is manually audited using efficient ballot sampling techniques. This strategy can achieve equal or greater confidence than precinct-based auditing at a significantly lower cost while protecting voter privacy better than previous ballot-based auditing methods. We show how to determine which ballots to audit against the recounting machines’ records and compare this new approach to precinct-based audits in the context of Virginia’s November 2006 election. Far fewer ballots need to be audited by hand using our approach. We also explore extensions to these techniques, such as varying individual ballots’ audit probabilities based on the votes they contain, that promise further efficiency gains.