The Witt group of a hyperelliptic curve over a field of characteristic different from two was determined by Parimala and Sujatha. Here, analogous results are obtained for the unramified Witt group in characteristic two using the analogue of Milnor's exact sequence for the Witt group of rational function fields developed earlier by the authors. In the elliptic case, if F is perfect and points of order two are rational, a generator and relation structure for the Witt group is given. (c) 2007 Elsevier Inc. All rights reserved.