Theoretical difficulties in describing the structure and thermodynamics of an ionic liquid double layer are often associated with the nonspherical shapes of ionic particles and extremely strong electrostatic interactions. The recent density functional theory predictions for the electrochemical properties of the double layer formed by a model ionic liquid wherein each cation is represented by two touching hard spheres, one positively charged and the other neutral, and each anion by a negatively charged hard spherical particle, remain untested in this strong coupling regime. We report results from a Monte Carlo simulation of this system. Because for an ionic liquid the Bjerrum length is exceedingly large, it is difficult to perform simulations under conditions of strong electrostatic coupling used in the previous density functional theory study. Results are obtained for a somewhat smaller (but still large) Bjerrum length so that reliable simulation data can be generated for a useful test of the corresponding theoretical predictions. On the whole, the density profiles predicted by the theory are quite good in comparison with the simulation data. The strong oscillations of ionic density profiles and the local electrostatic potential predicted by this theory are confirmed by simulation, although for a small electrode charge and strong electrostatic coupling, the theory predicts the contact ionic densities to be noticeably different from the Monte Carlo results. The theoretical results for the more important electrostatic potential profile at contact are given with good accuracy.