Inspired by the circle fitting algorithm "Hyper" of Al- Sharadqah and Chernov (1), which eliminates the second order "essential bias" that excludes terms of O(� 2 /N 2 ), we extend their analysis and show that by a small modification the second order bias can be eliminated com- pletely. By numerical experiments, we show that this results in better per- formance when the number N of points is small and the noise is large. AMS 2000 subject classifications: Primary 68T10, 68K45; secondary 68K40.