  Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve contained
in the projective plane. The Coolidge-Nagata conjecture asserts that $E$ is
Cremona equivalent to a line, i.e. it is mapped onto a line by some birational
transformation of $\mathbb{P}^2$. In arXiv:1405.5917 the second author analyzed
the log minimal model program run for the pair $(X,\frac{1}{2}D)$, where
$(X,D)\to (\mathbb{P}^2,E)$ is a minimal resolution of singularities, and as a
corollary he established the conjecture in case when more than one irreducible
curve in $\mathbb{P}^2\setminus E$ is contracted by the process of
minimalization. We prove the conjecture in the remaining cases.
