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Context.Stars form in the cold dense cores of interstellar molecular clouds and the detailed knowledge of the spectrum of masses of such cores is clearly a key for the understanding of the origin of the IMF. To date, observations have presented somewhat contradictory evidence relating to this issue. Aims.In this paper we propose to derive the mass function of a complete sample of dense molecular cores in a single cloud employing a robust method that uses uses extinction of background starlight to measure core masses and enables the reliable extension of such measurements to lower masses than previously possible. Methods.We use a map of near-infrared extinction in the nearby Pipe dark cloud to identify the population of dense cores in the cloud and measure their masses. Results.We identify 159 dense cores and construct the mass function for this population. We present the first robust evidence for a departure from a single power-law form in the mass function of a population of cores and find that this mass function is surprisingly similar in shape to the stellar IMF but scaled to a higher mass by a factor of about 3. This suggests that the distribution of stellar birth masses (IMF) is the direct product of the dense core mass function and a uniform star formation efficiency of 30%±10%, and that the stellar IMF may already be fixed during or before the earliest stages of core evolution. These results are consistent with previous dust continuum studies which suggested that the IMF directly originates from the core mass function. The typical density of ~104 cm-3 measured for the dense cores in this cloud suggests that the mass scale that characterizes the dense core mass function may be the result of a simple process of thermal (Jeans) fragmentation.