  We construct a coupling between a massive GFF and a random curve in which the
curve can be interpreted as the level line of the field and has the law of
massive SLE$_4$. This coupling is obtained by reweighting the law of the
standard coupling GFF-SLE$_4$ and our result can be seen as a conditional
version of the path-integral formulation of the massive GFF. We then show that
by reweighting the law of the coupling GFF-CLE$_4$ in a similar way, one
obtains a coupling between a massive GFF and a random countable collection of
simple loops, that we call massive CLE$_4$. Using this coupling, we relate
massive CLE$_4$ to the massive Brownian loop soup with intensity $1/2$, thus
proving a conjecture of Camia. As the law of the massive GFF, the laws of
massive SLE$_4$ and massive CLE$_4$ are conformally covariant.
