  We study the law of a hypoelliptic Brownian motion on an infinite-dimensional
Heisenberg group based on an abstract Wiener space. We show that the endpoint
distribution, which can be seen as a heat kernel measure, is absolutely
continuous with respect to a certain product of Gaussian and Lebesgue measures,
that the heat kernel is quasi-invariant under translation by the Cameron-Martin
subgroup, and that the Radon-Nikodym derivative is Malliavin smooth.
