We present X-ray and radio observations of the new Galactic supernova remnant (SNR) G306.3-0.9, recently discovered by Swift. Chandra imaging reveals a complex morphology, dominated by a bright shock. The X-ray spectrum is broadly consistent with a young SNR in the Sedov phase, implying an age of 2500 yr for a distance of 8 kpc, plausibly identifying this as one of the 20 youngest Galactic SNRs. Australia Telescope Compact Array imaging reveals a prominent ridge of radio emission that correlates with the X-ray emission. We find a flux density of ~160 mJy at 1 GHz, which is the lowest radio flux recorded for a Galactic SNR to date. The remnant is also detected at 24 mu m, indicating the presence of irradiated warm dust. The data reveal no compelling evidence for the presence of a compact stellar remnant.