I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a *unique* conformal structure on my new surface that is compatible with the conformal structures I started with.



Conformal Welding Reference
See MR1966191 (2005e:30012) 
Hamilton, D. H.(1-MD)
Conformal welding. Handbook of complex analysis: geometric function theory, Vol. 1, 137–146, North-Holland, Amsterdam, 2002. 
30C35 


and other papers by David Hamilton.


