  Given a Brownian motion $B_t$ and a general target law $\mu$ (not necessarily
centered or even integrable) we show how to construct an embedding of $\mu$ in
$B$. This embedding is an extension of an embedding due to Perkins, and is
optimal in the sense that it simultaneously minimises the distribution of the
maximum and maximises the distribution of the minimum among all embeddings of
$\mu$. The embedding is then applied to regular diffusions, and used to
characterise the target laws for which a $H^p$-embedding may be found.
