Assume we have a Riemann surface, the underlying topological surface of which is a sphere with (possibly uncountably many) points removed. Can we always conformally embed this Riemann surface in the Riemann sphere? If not, can someone suggest a counter example?

See e.g. here: Theorem 3.2.7. Any planar connected Riemann surface is biholomorphic to an open subset of $S^2$. The proof is very straightforward: Exhaust a genus $0$ surface $S$ by relatively compact domains $D_n$ each of which necessarily has genus $0$. For each $D_n$ find a conformal embedding $f_n$ to $S^2$. Now, normalize the family of mappings $f_n$ to to send a point $x\in D_1$ to a fixed point $z\in {\mathbb C}$ and to have unit derivative (in a chart) at $x$. Then use normality of the family of maps $f_n$ to get the limit (for a subsequence). Lastly, check that the limit is injective. This is the same argument Caratheodory used in his proof of uniformization theorem. 


Unless I am confused, why not add the points back, map the resulting surface to the Riemann sphere conformally by unifomization, then remove the images of the offending points? 

