  Let $G$ be the first Grigorchuk group. We show that the commutator width of
$G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and
also of six conjugates of $a$. Furthermore, we show that every finitely
generated subgroup $H\leq G$ has finite commutator width, which however can be
arbitrarily large, and that $G$ contains a subgroup of infinite commutator
width. The proofs were assisted by the computer algebra system GAP.
