  Methods used to detect giant exoplanets can be broadly divided into two
categories: indirect and direct. Indirect methods are more sensitive to planets
with a small orbital period, whereas direct detection is more sensitive to
planets orbiting at a large distance from their host star. %, and thus on long
orbital period. This dichotomy makes it difficult to combine the two techniques
on a single target at once. Simultaneous measurements made by direct and
indirect techniques offer the possibility of determining the mass and
luminosity of planets and a method of testing formation models. Here, we aim to
show how long-baseline interferometric observations guided by radial-velocity
can be used in such a way. We observed the recently-discovered giant planet
$\beta$ Pictoris c with GRAVITY, mounted on the Very Large Telescope
Interferometer (VLTI). This study constitutes the first direct confirmation of
a planet discovered through radial velocity. We find that the planet has a
temperature of $T = 1250\pm50$\,K and a dynamical mass of $M =
8.2\pm0.8\,M_{\rm Jup}$. At $18.5\pm2.5$\,Myr, this puts $\beta$ Pic c close to
a 'hot start' track, which is usually associated with formation via disk
instability. Conversely, the planet orbits at a distance of 2.7\,au, which is
too close for disk instability to occur. The low apparent magnitude ($M_{\rm K}
= 14.3 \pm 0.1$) favours a core accretion scenario. We suggest that this
apparent contradiction is a sign of hot core accretion, for example, due to the
mass of the planetary core or the existence of a high-temperature accretion
shock during formation.
