This paper presents a new method for fitting an ellipse to a point sequence extracted from images. It is widely known that the best fit is obtained by maximum likelihood. However, it requires iterations, which may not converge in the presence of large noise. Our approach is algebraic distance minimization; no iterations are required. Exploiting the fact that the solution depends on the way the scale is normalized, we analyze the accuracy to high order error terms with the scale normalization weight unspecified and determine it so that the bias is zero up to the second order. We demonstrate by experiments that our method is superior to the Taubin method, also algebraic and known to be highly accurate.