Marine Protected Areas (MPAs) are regions in the ocean or along coastlines where fishing is controlled to avoid the reduction or elimination of fish populations. A central question is where exactly to establish an MPA. We cast this as an optimal problem along a one-dimensional coast-line, where fish are assumed to move diffusively, and are subject to recruitment, natural death and harvesting through fishing. The functional being maximized is a weighted sum of the average fish density and the average fishing yield. It is shown that optimal controls exist, and that the location of the MPA is determined by two key model parameters, namely the size of the coast, and the weight of the average fish density in the functional.