  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 < 40pc (< 12mas angular separation, at 95%
confidence). We detect consistent angular broadening of the bursts and
persistent radio source (~ 2-4mas at 1.7GHz), which are both similar to the
expected Milky Way scattering contribution. The persistent radio source has a
projected size constrained to be < 0.7pc (< 0.2mas angular extent at 5.0GHz)
and a lower limit for the brightness temperature of T_b > 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.
