In this paper, we present a novel approach, called the "one-circle" algorithm, for measuring the eye gaze using a monocular image that zooms in on only one eye of a person. Observing that the iris contour is a circle, we estimate the normal direction of this iris circle, considered as the eye gaze, from its elliptical image. From basic projective geometry, an ellipse can be back-projected into space onto two circles of different orientations. However, by using an anthropometric property of the eyeball, the correct solution can be disambiguated. This allows us to obtain a higher resolution image of the iris with a zoom-in camera and thereby achieving higher accuracies in the estimation. The robustness of our gaze determination approach was verified statistically by the extensive experiments on synthetic and real image data. The two key contributions in this paper are that we show the possibility of finding the unique eye gaze direction from a single image of one eye and that one can obtain better accuracy as a consequence of this.