  A new explicit construction of Cauchy-Fantappi\'e kernels is introduced for
an arbitrary weakly pseudoconvex domain with smooth boundary. While not
holomorphic in the parameter, the new kernel reflects the complex geometry and
the Levi form of the boundary. Some estimates are obtained for the
corresponding integral operator, which provide evidence that this kernel and
related constructions give useful new tools for complex analysis on this
general class of domains.
