We have studied the correlation of the 1/f noise level in an RF SQUID, at frequencies from 1 Hz down to below.001 Hz, with the amount of trapped magnetic flux in the SQUID structure and the magnitude of its temperature fluctuations. Trapped flux levels corresponding to cooldown fields from 0.6 G to less than 10-7 G were obtained through the use of appropriate magnetic shields coupled with a special feedback-controlled cooldown technique designed to minimize the trapping of thermoelectrically generated flux. Temperature fluctuations were measured with high resolution thermometry and were controlled in some cases to 10 µK. Results show decreased low frequency 1/f noise under conditions of low trapped flux and highly-stabilized SQUID temperature.