  We construct a conformally invariant random family of closed curves in the
plane by welding of random homeomorphisms of the unit circle. The homeomorphism
is constructed using the exponential of $\beta X$ where $X$ is the restriction
of the two dimensional free field on the circle and the parameter $\beta$ is in
the "high temperature" regime $\beta<\sqrt 2$. The welding problem is solved by
studying a non-uniformly elliptic Beltrami equation with a random complex
dilatation. For the existence a method of Lehto is used. This requires sharp
probabilistic estimates to control conformal moduli of annuli and they are
proven by decomposing the free field as a sum of independent fixed scale fields
and controlling the correlations of the complex dilation restricted to dyadic
cells of various scales. For uniqueness we invoke a result by Jones and Smirnov
on conformal removability of H\"older curves. We conjecture that our curves are
locally related to SLE$(\kappa)$ for $\kappa<4$.
