We present a current-controlled magnonic crystal consisting of a ferrite film in which spin waves propagate and a set of parallel, periodically spaced, current conducting stripes placed close to the film surface. The current flow causes a sine-like variation of the film's internal magnetic field, which can be modulated by changing the amount of current. Transmission measurements reveal a single, pronounced rejection band. With increasing current strength the rejection band depth and its width increase strongly. Moreover, it is possible to switch the artificial, periodic structure on and off, so that the waveguide makes a transition from full rejection to full transmission within less than 50 ns. Numerical simulations confirm the experimental results and show that the spin-wave propagation in the crystal can be effectively described as a scattering process in the first Born approximation. Three ways to increase the reflection efficiency of the magnonic crystal are identified: an increased number of periods, an increased lattice constant and a decreased spacing between the current carrying structure and the waveguide.