We present an analysis of the mass distribution inferred from strong lensing by SPT-CL J0356-5337, a cluster of galaxies at redshift $z=1.0359$ revealed in the follow-up of the SPT-SZ clusters. The cluster has an Einstein radius of ${\theta }_{{\rm{E}}}\simeq $ 14″ for a source at z = 3 and a mass within 500 kpc of ${M}_{500\mathrm{kpc}}=4.0\pm 0.8\times {10}^{14}\,$ ${M}_{\odot }$ . Our spectroscopic identification of three multiply imaged systems ( $z=2.363$ , $z=2.364$ , and $z=3.048$), combined with HSTF606W-band imaging allows us to build a strong lensing model for this cluster with an rms of $\leqslant 0\buildrel{\prime\prime}\over{.} 3$ . Our modeling reveals a two- component mass distribution in the cluster. One mass component is dominated by the brightest cluster Galaxy and the other component, separated by ∼170 kpc, contains a group of eight red elliptical galaxies confined in a ∼9″ (∼70 kpc) diameter circle. We estimate the mass ratio between the two components to be between 1:1.25 and 1:1.58. In addition, spectroscopic data reveal that these two near-equal mass cores have only a small velocity difference of ∼300 km s-1 between the two components. This small radial velocity difference suggests that most of the relative velocity takes place in the plane of the sky, and implies that SPT-CL J0356-5337 is a major merger with a small impact parameter seen face-on. We also assess the relative contributions of Galaxy-scale halos to the overall mass of the core of the cluster and find that within 800 kpc from the brightest cluster Galaxy about 27% of the total mass can be attributed to visible and dark matter associated with galaxies, whereas only 73% of the total mass in the core comes from cluster-scale dark matter halos.
