This paper considers a boundary feedback control problem for two first-order, nonlinearly coupled, hyperbolic partial differential equations with Lotka-Volterra type coupling. Boundary control action is used on one equation to drive the state at the end of the spatial domain to a desired constant reference value. Static and dynamic boundary controllers are designed based on a special Lyapunov functional that is related to an entropy function. The time derivative of the entropy function is made strictly negative by an appropriate choice of boundary conditions. A unique classical solution is shown to exist globally in time and (asymptotic) exponential convergence to the desired steady-state solution is shown in the (C-0) L-2-norm. The boundary control design is illustrated with simulations.