Two major approaches have been proposed to efficiently process queries in databases: Speeding up the search by using index structures, and speeding up the search by operating on a compressed database, such as a signature file. Both approaches have their limitations: Indexing techniques are inefficient in extreme configurations, such as high-dimensional spaces, where even a simple scan may be cheaper than an index-based search. Compression techniques are not very efficient in all other situations. We propose to combine both techniques to search for nearest neighbors in a high-dimensional space.For this purpose, we develop a compressed index, called the IQ-tree, with a three-level structure: The first level is a regular (flat) directory consisting of minimum bounding boxes, the second level contains data points in a compressed representation, and the third level contains the actual data.We overcome several engineering challenges in constructing an effective index structure of this type. The most significant of these is to decide how much to compress at the second level. Too much compression will lead to many needless expensive accesses to the third level. Too little compression will increase both the storage and the access cost for the first two levels.We develop a cost model and an optimization algorithm based on this cost model that permits an independent determination of the degree of compression for each second level page to minimize expected query cost. In an experimental evaluation, we demonstrate that the IQ-tree shows a performance that is the "best of both worlds" for a wide range of data distributions and dimensionalities.