  In the theory of reproducing kernel Hilbert spaces, weak product spaces
generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick
spaces $\mathcal H$, we characterize all multipliers of the weak product space
$\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has
the so-called column-row property, then the multipliers of $\mathcal H$ and of
$\mathcal H \odot \mathcal H$ coincide. This result applies in particular to
the classical Dirichlet space and to the Drury-Arveson space on a finite
dimensional ball. As a key device, we exhibit a natural operator space
structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations
of completely bounded maps.
