  We present the most sensitive direct imaging and radial velocity (RV)
exploration of $\epsilon$ Eridani to date. $\epsilon$ Eridani is an adolescent
planetary system, reminiscent of the early Solar system. It is surrounded by a
prominent and complex debris disk which is likely stirred by one or several gas
giant exoplanets. The discovery of the RV signature of a giant exoplanet was
announced 15 years ago, but has met with scrutiny due to possible confusion
with stellar noise. We confirm the planet with a new compilation and analysis
of precise RV data spanning 30 years, and combine it with upper limits from our
direct imaging search, the most sensitive ever performed. The deep images were
taken in the Ms band (4.7$\mu$m) with the vortex coronagraph recently installed
in W.M. Keck Observatory's infrared camera NIRC2, which opens a sensitive
window for planet searches around nearby adolescent systems. The RV data and
direct imaging upper limit maps were combined in an innovative joint Bayesian
analysis, providing new constraints on the mass and orbital parameters of the
elusive planet. $\epsilon$ Eridani b has a mass of $0.78^{+0.38}_{-0.12}$
$M_{Jup}$ and is orbiting $\epsilon$ Eridani at about $3.48\pm 0.02$ AU with a
period of $7.37 \pm 0.07$ years. The eccentricity of $\epsilon$ Eridani b's
orbit is $0.07^{+0.06}_{-0.05}$, an order of magnitude smaller than early
estimates and consistent with a circular orbit. We discuss our findings from
the standpoint of planet-disk interactions and prospects for future detection
and characterization with the James Webb Space Telescope.
