The millisecond-duration radio flashes known as Fast Radio Bursts (FRBs) represent an enigmatic astrophysical phenomenon. Recently, the sub-arcsecond localization (~ 100mas precision) of FRB121102 using the VLA has led to its unambiguous association with persistent radio and optical counterparts, and to the identification of its host galaxy. However, an even more precise localization is needed in order to probe the direct physical relationship between the millisecond bursts themselves and the associated persistent emission. Here we report very-long-baseline radio interferometric observations using the European VLBI Network and the 305-m Arecibo telescope, which simultaneously detect both the bursts and the persistent radio emission at milliarcsecond angular scales and show that they are co-located to within a projected linear separation of (( 5 x 10^7K. Together, these observations provide strong evidence for a direct physical link between FRB121102 and the compact persistent radio source. We argue that a burst source associated with a low-luminosity active galactic nucleus or a young neutron star energizing a supernova remnant are the two scenarios for FRB121102 that best match the observed data.