We demonstrate the passive control of viscous flow in a channel by using an elastic arch embedded in the flow. Depending on the fluid flux, the arch may "snap" between two states-constricting and unconstricting-that differ in hydraulic conductivity by up to an order of magnitude. We use a combination of experiments at a macroscopic scale and theory to study the constricting and unconstricting states, and determine the critical flux required to transition between them. We show that such a device may be precisely tuned for use in a range of applications, and, in particular, has potential as a passive microfluidic fuse to prevent excessive fluxes in rigid-walled channels.