  We consider the Schramm-Loewner evolution (SLE$_\kappa$) with $\kappa=4$, the
critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve
and above which it is self-intersecting. We show that the range of an SLE$_4$
curve is a.s. conformally removable, answering a question posed by Sheffield.
Such curves arise as the conformal welding of a pair of independent critical
($\gamma=2$) Liouville quantum gravity (LQG) surfaces along their boundaries
and our result implies that this conformal welding is unique. In order to
establish this result, we give a new sufficient condition for a set $X
\subseteq {\mathbf C}$ to be conformally removable which applies in the case
that $X$ is not necessarily the boundary of a simply connected domain.
