This paper continues the development of a stochastic calculus for two-parameter martingales, and particularly for the two-parameter Wiener process. Whereas in an earlier paper we showed that two types of stochastic integrals were necessary for representing functionals and martingales of a Wiener process, introduction of two mixed area integrals is necessary to complete the stochastic calculus. These mixed integrals are weak martingales in the sense of Cairoli and Walsh, and are necessary in a general representation for weak martingales and transformations of weak martingales. Stopping times are introduced for two-parameter processes, and a characterization of strong martingales in terms of stopping times is given.