In this paper, we develop and analyze a gossip-based average consensus algorithm that enables all of the components of a distributed system, each with some initial value, to reach (approximate) average consensus on their initial values after executing a finite number of iterations, and without having to reveal to other curious components the specific value they contribute to the average calculation. We consider a fully-connected (undirected) network in which curious components do not interfere in the computation in any other way, but can collaborate arbitrarily and are aware of the privacy-preserving strategy. We characterize precisely conditions on the information exchange that guarantee privacy preservation for a specific node. The protocol also provides a criterion that allows the nodes to determine, in a distributed manner (while running the proposed gossip protocol), when to terminate their operation because approximate average consensus has been reached.