To understand ice nucleation under high supersaturation in clouds, measurements of natural ice nuclei for supersaturation with respect to water (Sw) up to 10% from -20 to -24C were carried out using a horizontal gradient, continuous flow, ice thermal diffusion chamber after solving the spurious count problem due to ice crystal protrusion on the bottom plate with the addition of a small amount of ethylene glycol. The measurements were made with continental air masses in early summer of 1993 at Salt Lake City, Utah in the United States of America.The measured ice nucleus concentration under the condition of -20C and Sw 5% showed a daily variation. The ice nucleus concentration changed from about 10l-1 in a warm air mass to about 1l-1 in a cold air mass, respectively, before and after the passage of a cold front in the synoptic scale. It was inferred that the instability of the warm air mass helps raise the particulates from or near the ground to increase the ice nucleus concentration. In order to pick out the supersaturation dependence at the various temperatures, the ratio of the ice nucleus concentration (N) to that at water saturation (No) was estimated. On average, Sw increases of 5 and 10% raised the N/No ratio by as much as a factor of two or more, respectively. It was found that a general relationship exists between N/No and Sw, and the slope further steepens above water saturation.