  We report a comprehensive spectro-polarimetric study of the black hole binary
LMC X$-3$ using simultaneous {\it IXPE}, {\it NICER} and {\it NuSTAR}
observations in 0.5$-$20 keV energy band. The broad-band energy spectrum
(0.5$-$20 keV) with {\it NICER} and {\it NuSTAR} is well described by the disc
emission of temperature $\sim$1.1 keV and a weak Comptonizing tail beyond
$\sim$10 keV. This evidently suggests a disc-dominated spectral state of the
source with disc contribution of $\sim$96\%. The lack of variability
($rms$$\sim$0.5\%) in the power spectrum further corroborates the high/soft
nature of the source. A significant polarization degree (PD) of 3.04$\pm$0.40\%
(>7$\sigma$) at a polarization angle (PA) of
$-$44.24$^{\circ}$$\pm$3.77$^{\circ}$ (>7$\sigma$) is found in 2$-$8 keV energy
range of {\it IXPE}. In addition, PD is seen to increase with energy up to
$\sim$4.35$\pm$0.98\% (>3$\sigma$) in 4$-$8 keV band. Further, we attempt to
constrain the source spin ($a_{*}$) using broad-band spectral modelling that
indicates a weakly rotating `hole' in LMC X$-3$ with
$a_{*}=0.273_{-0.012}^{+0.011}-0.295_{-0.021}^{+0.008}$ ($90\%$ confidence).
Based on the spectro-polarimetric results, we infer that the polarization in
LMC X$-3$ is resulted possibly due to the combined effects of the direct and/or
reflected emissions from a partially ionized disc atmosphere. Finally, we
discuss the relevance of our findings.
