We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $\varphi \colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $\varphi(η) = \tildeη$ then $\varphi$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $\varphi$ is a homeomorphism which is conformal on ${\mathbf H} \setminus η$ and $\varphi(η)$, $η$ are equal in distribution then $\varphi$ is a conformal automorphism of ${\mathbf H}$. Applying this result for $κ=4$ establishes that the welding operation for critical ($γ=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $κ\in (4,8)$ gives a new proof that the welding of two independent $κ/4$-stable looptrees of quantum disks to produce an SLE$_κ$ on top of an independent $4/\sqrtκ$-LQG surface is well-defined.