  Let $E\subseteq \mathbb{P}^2$ be a complex curve homeomorphic to the
projective line. The Negativity Conjecture asserts that the Kodaira-Iitaka
dimension of $K_X+\frac{1}{2}D$, where $(X,D)\to (\mathbb{P}^{2},E)$ is a
minimal log resolution, is negative. We prove structure theorems for curves
satisfying this conjecture and we finish their classification up to a
projective equivalence by describing the ones whose complement admits no
$\mathbb{C}^{**}$-fibration. As a consequence, we show that they satisfy the
Strong Rigidity Conjecture of Flenner-Zaidenberg. The proofs are based on the
almost minimal model program. The obtained list contains one new series of
bicuspidal curves.
