A theoretical Study is presented on the shape of the interface between coexisting isotropic and nematic phases in contact with a solid vertical wall. The interface profile is determined by a competition between three surface tensions, two anchoring strengths, gravity, and the Frank elastic constants of the director field. In the weak-anchoring limit, the director field is rigid and uniform, and we find the capillary rise height to depend nontrivially on the orientation of the director field relative to the solid-fluid interface. For strong surface anchoring, the director field adjusts to the preferred homeotropic or planar anchoring at the solid-liquid and liquid-liquid interfaces. The shape of die interfacial profile is now a function of the balance between the surface energy and the splay and bend elastic deformation energies. Interestingly, for both weak and strong anchoring the profile decays nonmonotonically albeit only very weakly so. We compare our theory with recent capillary rise experiments on coexisting isotropic and nematic phases of colloidal platelets [van der Beek et al. Phys. Rev. Lett. 2006, 97, 087801] and are able to extract from the experimental data the surface tension and the anchoring strength.