  We consider a Stratonovich heat equation in $(0,1)$ with a nonlinear
multiplicative noise driven by a trace-class Wiener process. First, the
equation is shown to have a unique mild solution. Secondly, convolutional rough
paths techniques are used to provide an almost sure continuity result for the
solution with respect to the solution of the 'smooth' equation obtained by
replacing the noise with an absolutely continuous process. This continuity
result is then exploited to prove weak convergence results based on Donsker and
Kac-Stroock type approximations of the noise.
