We report on the design, fabrication, and measurement of a low-temperature microwave switch architecture that can be fully integrated on chip using single Josephson junctions and superconducting bias lines. The basic switching element used is a reflective single-pole-single-throw switch that is realized by tuning the critical current of a single Josephson junction by way of a local flux bias. A single-pole-single-throw switch is demonstrated at 4.2 K to have an on/off ratio in excess of 20 dB up to 12 GHz. This element is then incorporated into the design of a single-pole-double-throw switch that exhibits an on/off ratio above 20 dB up to 10 GHz. The switches require no intrinsic power dissipation in either state, and their performance was verified for input powers of -100 dBm to -60 dBm. S-parameter simulations using a simple lumped element model are in good agreement with the experimental data, allowing for the design to be extended to larger switch architectures.