The first four strings of phototubes for the AMANDA high-energy neutrino observatory are now frozen in place at a depth of 800 to 1000 m in ice at the South Pole. During the 1995-96 season an additional six strings will be deployed at greater depths. Provided absorption, scattering, and refraction of visible light are sufficiently small, the trajectory of a muon into which a neutrino converts can be determined by using the array of phototubes to measure the arrival times of \v{C}erenkov light emitted by the muon. To help in deciding on the depth for implantation of the six new strings, we discuss models of age vs depth for South Pole ice, we estimate mean free paths for scattering from bubbles and dust as a function of depth, and we assess distortion of light paths due to refraction at crystal boundaries and interfaces between air-hydrate inclusions and normal ice. We conclude that the depth interval 1600 to 1800 m will be suitably transparent for the next six AMANDA strings and, moreover, that the interval 1600 to 2100 m will be suitably transparent for a future 1-km 3   observatory except possibly in a region a few tens of meters thick at a depth corresponding to a peak in the dust concentration at 60 kyr BP.