We develop a mathematical optimization model at the intersection of homeland security and immigration, which chooses various immigration enforcement decision variables to minimize the probability that a terrorist can successfully enter the U.S. across the U.S.-Mexico border. Included are a discrete choice model for the probability that a potential alien crosser will attempt to cross the U.S.-Mexico border in terms of the likelihood of success and the U.S. wage for illegal workers, a spatial model that calculates the apprehension probability as a function of the number of crossers, the number of border patrol agents and the amount of surveillance technology on the border, a queueing model that determines the probability that an apprehended alien will be detained and removed as a function of the number of detention beds, and an equilibrium model for the illegal wage, which balances the supply and demand for work and incorporates the impact of worksite enforcement. Our main result is that detention beds are the current system bottleneck (even after the large reduction in detention residence times recently achieved by Expedited Removal), and increases in border patrol staffing or surveillance technology would not provide any improvements without a large increase in detention capacity. Our model also predicts that surveillance technology is more cost-effective than border patrol agents, which in turn are more cost-effective than worksite inspectors, but these results are not robust due to the difficulty of predicting human behavior from existing data. Overall, the probability that a terrorist can successfully enter the U.S. is very high, and it would be extremely costly and difficult to significantly reduce it. We also investigate the alternative objective function of minimizing the flow of illegal aliens across the U.S.-Mexico border, and obtain qualitatively similar results.