We have investigated the effects of shear flow on a polymeric bicontinuous microemulsion using neutron scattering, light scattering, optical microscopy, and rheology. The microemulsion consists of a ternary blend of poly(ethyl ethylene) (PEE), poly(dimethyl siloxane) (PDMS), and a PEE-PDMS diblock copolymer. At equilibrium, the microemulsion contains two percolating microphases, one PEE rich and the other PDMS rich, separated by a copolymer-laden interface; the characteristic length scale of this structure is 80 nm. Low strain amplitude oscillatory shear measurements reveal behavior similar to that of block copolymer lamellar phases just above the order-disorder transition. Steady shear experiments expose four distinct regimes of response as a function of the shear rate. At low shear rates (regime I) Newtonian behavior is observed, whereas at intermediate shear rates (regime II) development of anisotropy in the morphology leads to shear thinning. When the shear rate is further increased, there is an abrupt breakdown of the bicontinuous structure, resulting in flow-induced phase separation (regime III). Rheological measurements indicate that the shear stress is almost independent of the shear rate in this regime. Light scattering reveals a streak-like pattern, and correspondingly a string-like morphology with micron dimensions is observed with video microscopy. Upon a further increase of the shear rate (regime IV), the sample resembles an immiscible binary polymer blend with the block copolymer playing no significant role; the stress increases strongly with the shear rate. In some respects these results resemble those from other weakly structured complex fluids (sponge phases, liquid crystals, worm-like micelles, block copolymers, polymer blends, and polymer solutions), yet in important ways this system is unique.