  We present the first joint NuSTAR and NICER observations of the ultra-compact
X-ray binary (UCXB) 4U 1543$-$624 obtained in 2020 April. The source was at a
luminosity of $L_{0.5-50\ \mathrm{keV}} = 4.9 (D/7\
\mathrm{kpc})^{2}\times10^{36}$ ergs s$^{-1}$ and showed evidence of reflected
emission in the form of an O VIII line, Fe K line, and Compton hump within the
spectrum. We used a full reflection model, known as xillverCO, that is tailored
for the atypical abundances found in UCXBs, to account for the reflected
emission. We tested the emission radii of the O and Fe line components and
conclude that they originate from a common disk radius in the innermost region
of the accretion disk ($R_{\rm in} \leq1.07\ R_{\mathrm{ISCO}}$). Assuming that
the compact accretor is a neutron star (NS) and the position of the inner disk
is the Alfv\'{e}n radius, we placed an upper limit on the magnetic field
strength to be $B\leq0.7(D/7\ \mathrm {kpc})\times10^{8}$ G at the poles. Given
the lack of pulsations detected and position of $R_{\rm in}$, it was likely
that a boundary layer region had formed between the NS surface and inner edge
of the accretion disk with an extent of 1.2 km. This implies a maximum radius
of the neutron star accretor of $R_{\mathrm{NS}}\leq 12.1$ km when assuming a
canonical NS mass of 1.4 $M_{\odot}$.
