Incomplete knowledge of the pattern of magnetic lineations and fossil transform faults represented by sea floor spreading data on two plates generated by the same spreading center leads to uncertainties in a reconstruction of the past relative configuration of the plates. A reconstruction may be represented by the finite rotation that describes one configuration relative to another. In this paper a method of reconstruction is provided that reflects the uncertainties in the data. The method minimizes a weighted least squares measure of fit as a function of the rotation parameters. For a given rotation the measure of fit represents the sum of squares of the weighted distances of fixed and rotated data points (representing corresponding former plate margin segments) from a common plate margin. The common margin is estimated from the locations of both fixed and rotated data points. The estimated margin consists of a set of great-circle arcs. It is shown how the method of reconstruction may be utilized to obtain an uncertainty region for the pole and angle of rotation that characterize the finite rotation. These techniques are then used to study the uncertainties of reconstructions in the South Pacific for the times of anomalies 13 and 18. The best fit reconstructions and the uncertainty region for the anomaly 18 pole are in good agreement with previous work. However, the uncertainties in the anomaly 13 pole were substantially underestimated by previous investigators.