Abell 2146 ($z$ = 0.232) consists of two galaxy clusters undergoing a major merger. The system was discovered in previous work, where two large shock fronts were detected using the $Chandra\; X-ray\; Observatory$, consistent with a merger close to the plane of the sky, caught soon after first core passage. A weak gravitational lensing analysis of the total gravitating mass in the system, using the distorted shapes of distant galaxies seen with Advanced Camera for Surveys - Wide Field Channel on $Hubble\; Space\; Telescope$, is presented. The highest peak in the reconstruction of the projected mass is centred on the brightest cluster galaxy (BCG) in Abell 2146-A. The mass associated with Abell 2146-B is more extended. Bootstrapped noise mass reconstructions show the mass peak in Abell 2146-A to be consistently centred on the BCG. Previous work showed that BCG-A appears to lag behind an X-ray cool core; although the peak of the mass reconstruction is centred on the BCG, it is also consistent with the X-ray peak given the resolution of the weak lensing mass map. The best-fitting mass model with two components centred on the BCGs yields $M200$ = 1.1$\mathrm{<mrow\; data-mjx-texclass="ORD">-0.4</mrow><mrow\; data-mjx-texclass="ORD">+0.3</mrow>}$ × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">15</mrow>}$ and 3$\mathrm{<mrow\; data-mjx-texclass="ORD">-2</mrow><mrow\; data-mjx-texclass="ORD">+1</mrow>}$ × 10$\mathrm{<mrow\; data-mjx-texclass="ORD">14</mrow>}$ M$\odot$ for Abell 2146-A and Abell 2146-B, respectively, assuming a mass concentration parameter of $c$ = 3.5 for each cluster. From the weak lensing analysis, Abell 2146-A is the primary halo component, and the origin of the apparent discrepancy with the X-ray analysis where Abell 2146-B is the primary halo is being assessed using simulations of the merger.