Positive systems represent a class of systems whose state variables are nonnegative. This paper presents an L1 adaptive controller for such systems. The control objective is to force the system output track a given nonnegative reference signal. In the analysis of the L1 adaptive controller, we first show the positivity of the constructed reference system, and then demonstrate that the controlled system states track those of the reference system. We provide the uniform transient and steady-state performance bounds on the system state and the control signals, which can be systematically improved by increasing the adaptation rate, under the assumption that the unknown system parameters are positive. Simulations are presented to corroborate our results.