Analysis of radio-metric tracking data from the Pioneer 10/11 spacecraft at distances between 20 and 70 astronomical units (AU) from the Sun has consistently indicated the presence of an anomalous, small, constant Doppler frequency drift. The drift can be interpreted as being due to a constant acceleration of a_P = (8.74 ± 1.33) × 10⁻⁸ cm s⁻² directed towards the Sun. Although it is suspected that there is a systematic origin to the effect, none has been found. As a result, the nature of this anomaly has become of growing interest. Here we present a concept for a deep-space experiment that will reveal the origin of the discovered anomaly and will also characterize its properties to an accuracy of at least two orders of magnitude below the anomaly's size. The proposed mission will not only provide a significant accuracy improvement in the search for small anomalous accelerations, it will also determine if the anomaly is due to some internal systematic or has an external origin. A number of critical requirements and design considerations for the mission are outlined and addressed. If only already existing technologies were used, the mission could be flown as early as 2010.