Contrary to expectations based on mode spacing, single-mode operation in very large He-Ne ring lasers may be achieved at intracavity power levels up to approximately0.15 times the saturation intensity for the He-Ne transition. Homogeneous line broadening at a high total gas pressure of 4-6 Torr allows a single-peaked gain profile that suppresses closely spaced multiple modes. At startup, decay of initial multiple modes may take tens of seconds. The single remaining mode in each direction persists metastably as the cavity is detuned by many times the mode frequency spacing. A theoretical explanation requires the gain profile to be concave down and to satisfy an inequality related to slope and saturation at the operating frequency. Calculated metastable frequency ranges are > 150 MHz at 6 Torr and depend strongly on pressure. Examples of unusual stable mode configurations are shown, with differently numbered modes in the two directions and with multiple modes at a spacing of approximately 100 MHz.