Currently there is no a official driver available for the HP Color LaserJet 1600, however there is a simple work-around.

First download and install the HP Printer Drivers v.2.12 for OS X - from Apple Support.

Next install the HP Color LaserJet 1600 using the “HP Color LaserJet 2600” from the dropdown list in the config section as the driver for the HP Color LaserJet 1600.

Confirmed to work for OS X Mountain Lion, OS X Lion and Mac OS X v10.6 Snow Leopard.