  We prove a compact version of the $T1$ theorem for bi-parameter singular
integrals. That is, if a bi-parameter singular integral operator $T$ admits the
compact full and partial kernel representations, and satisfies the weak
compactness property, the diagonal $\mathrm{CMO}$ condition, and the product
$\mathrm{CMO}$ condition, then $T$ can be extended to a compact operator on
$L^p(w)$ for all $1<p<\infty$ and $w \in A_p(\mathbb{R}^{n_1} \times
\mathbb{R}^{n_2})$. Even in the unweighted setting, it is the first time to
give a compact extension of Journ\'{e}'s $T1$ theorem on product spaces.
