We present the publicly available moving-mesh hydrodynamics code Sprout. Sprout solves the equations of ideal hydrodynamics on an expanding Cartesian mesh. The expanding mesh can follow fluid outflows for several orders of magnitude with very little numerical diffusion, thereby capturing shocks and fine structures accurately. Following the bulk flow accurately also allows for longer timesteps in general. This makes Sprout particularly suitable for studying expanding outflows such as supernova remnants and active galactic nuclei. Relative to other moving mesh codes, the simple mesh structure in Sprout is also convenient for implementing additional physics or algorithms. Many code tests are performed to test the accuracy and performance of the numerical scheme.