  Given a finitely connected region $\Omega$ of the Riemann sphere whose
complement consists of $m$ mutually disjoint closed disks $\bar{U}_j$, the
random homeomorphism $h_j$ on the boundary component $\partial U_j$ is
constructed using the exponential Gaussian free field. The existence and
uniqueness of random conformal welding of $\Omega$ with $h_j$ is established by
investigating a non-uniformly elliptic Betrami equation with a random complex
dilatation. This generalizes the result of Astala, Jones, Kupiainen and Saksman
to multiply connected domains.
