  We present 850 $\mu$m imaging polarimetry data of the $\rho$ Oph-A core taken
with the Submillimeter Common-User Bolometer Array-2 (SCUBA-2) and its
polarimeter (POL-2), as part of our ongoing survey project, BISTRO (B-fields In
STar forming RegiOns). The polarization vectors are used to identify the
orientation of the magnetic field projected on the plane of the sky at a
resolution of 0.01 pc. We identify 10 subregions with distinct polarization
fractions and angles in the 0.2 pc $\rho$ Oph A core; some of them can be part
of a coherent magnetic field structure in the $\rho$ Oph region. The results
are consistent with previous observations of the brightest regions of $\rho$
Oph-A, where the degrees of polarization are at a level of a few percents, but
our data reveal for the first time the magnetic field structures in the fainter
regions surrounding the core where the degree of polarization is much higher
($> 5 \%$). A comparison with previous near-infrared polarimetric data shows
that there are several magnetic field components which are consistent at
near-infrared and submillimeter wavelengths. Using the
Davis-Chandrasekhar-Fermi method, we also derive magnetic field strengths in
several sub-core regions, which range from approximately 0.2 to 5 mG. We also
find a correlation between the magnetic field orientations projected on the sky
with the core centroid velocity components.
