Isogeometric analysis (IGA) is a novel discretization method, introduced by Hughes, which is based on nonuniform rational B-splines (NURBS). Among other features, IGA uses directly the geometry description coming from computer-aided design software without approximation, and the analysis is performed using shape functions of variable (possibly high) regularity. In this paper we propose a new discretization scheme based on continuous B-splines, adapting the IGA to the solution of Maxwell's equations. We present extensive numerical results to show that our scheme is free of spurious modes, and that it approximates singular solutions in domains with reentrant corners and edges.