Circular markers are planar markers which offer great performances for detection and pose estimation. For an uncalibrated camera with an unknown focal length, the images of at least two coplanar circles are generally required to recover their poses and their supporting plane. Unfortunately, detecting more than one ellipse in the image is tricky and time-consuming, especially regarding concentric circles. On the other hand, when the camera is calibrated, the pose of one circle can be computed with its image alone but the solution is twofold and can hardly be disambiguated. Our contribution is to put beyond this limit by dealing with the uncalibrated case of a camera seeing one circle and highlighting geometric configuration where the ambiguity can be analysed. We propose a new problem formulation that enables to show how to detect geometric configurations in which the ambiguity can be removed. Furthermore, we introduce the notion of default camera intrinsics and show, empirically, the surprising observation that very approximate calibration leads to a circle pose estimation which gives accurate results in reprojection for augmented reality.