  It has long been understood that non-trivial Conformal Field Theories (CFTs)
with vanishing central charge ($c=0$) are logarithmic. So far however, the
structure of the identity module -- the (left and right) Virasoro descendants
of the identity field -- had not been elucidated beyond the stress-energy
tensor $T$ and its logarithmic partner $t$ (the solution of the "$c\to 0$
catastrophe"). In this paper, we determine this structure together with the
associated OPE of primary fields up to level $h=\bar{h}=2$ for polymers and
percolation CFTs. This is done by taking the $c\to 0$ limit of $O(n)$ and Potts
models and combining recent results from the bootstrap with arguments based on
conformal invariance and self-duality. We find that the structure contains a
rank-3 Jordan cell involving the field $T\bar{T}$, and is identical for
polymers and percolation. It is characterized in part by the common value of a
non-chiral logarithmic coupling $a_0=-{25\over 48}$.
