  $J/\psi$ suppression has long been considered a sensitive signature of the
formation of the Quark-Gluon Plasma (QGP) in relativistic heavy-ion collisions.
In this letter, we present the first measurement of inclusive $J/\psi$
production at mid-rapidity through the dimuon decay channel in Au+Au collisions
at $\sqrt{s_{NN}}$ = 200 GeV with the STAR experiment. These measurements
became possible after the installation of the Muon Telescope Detector was
completed in 2014. The $J/\psi$ yields are measured in a wide transverse
momentum ($p_{\rm{T}}$) range of 0.15 GeV/$c$ to 12 GeV/$c$ from central to
peripheral collisions. They extend the kinematic reach of previous measurements
at RHIC with improved precision. In the 0-10% most central collisions, the
$J/\psi$ yield is suppressed by a factor of approximately 3 for $p_{\rm{T}}>5$
GeV/$c$ relative to that in p+p collisions scaled by the number of binary
nucleon-nucleon collisions. The $J/\psi$ nuclear modification factor displays
little dependence on $p_{\rm{T}}$ in all centrality bins. Model calculations
can qualitatively describe the data, providing further evidence for the
color-screening effect experienced by $J/\psi$ mesons in the QGP.
