A new structural phase transition has been observed at low temperature for the one-third of a monolayer (alpha phase) of Pb on Ge(111) using a variable-temperature scanning tunneling microscope. The well-known (sqrt[3] x sqrt[3])R30 degrees to (3 x 3) transition is accompanied by a new structural phase transition from (3 x 3) to a disordered phase at approximately 76 K. The formation of this "glasslike" phase is a consequence of competing interactions on different length scales.