 Solution

The Riemann sphere has three complex Conformal Killing vector fields, forming the Möbius group $PSL(2,\mathbb{C})$. Gauge fixing this residual conformal symmetry permits three insertion points to be fixed arbitrarily. Each unintegrated string vertex operator contains $c\tilde{c}$, and the three insertions exactly saturate the three holomorphic and three antiholomorphic ghost zero modes. Since every full vertex has weight $(0,0)$, the correlator is Möbius invariant and independent of the chosen three positions.