Observations with the Hubble Space Telescope reveal an irregular network of dust spiral arms in the nuclear region of the interacting disk galaxy NGC 2207. The spirals extend from ~50 pc to ~300 pc in galactocentric radius, with a projected width of ~20 pc. Radiative transfer calculations determine the gas properties of the spirals and the inner disk, and imply a factor of ~4 local gas compression in the spirals. The gas is not strongly self-gravitating, nor is there a nuclear bar, so the spirals could not have formed by the usual mechanisms applied to main galaxy disks. Instead, they may result from acoustic instabilities that amplify at small galactic radii. Such instabilities may promote gas accretion into the nucleus.