Reference for Riemann Existence Theorem I am looking for a proof of the following result: 
Theorem Let $Y$ be a [connected] Riemann surface, $S$ a discrete subset of $Y$ and $p:X' \to Y-S$ a finite sheeted [connected] covering of $Y$. Then there is a Riemann surface $X$, unique up to biholomorphism, together with an embedding $X' \hookrightarrow X$ such that $X-X'$ is discrete and $p$ extends to a proper holomorphic mapping $\tilde p:X \to Y$. 
I know there are very different version of this theorem, but I'm looking for this one in particular. The most close I've found in literature is in Fulton's Algebraic Topology, but just a weaker version (S finite) is proven and most statements are left as exercises. 
So far I've looked on Miranda (only deals with S finite), Donaldson (deals with S infinte and discrete, but leaves most of the statements unproven), Forster and Farkas (totally lack this result); I'm also aware of this similar request, but it is about the "S finite" case and anyway I was not able to recover the paper suggested in the answer. 
 A: I was misremembering: The related questions here and here while address similar issues do not answer your question. 


*

*Connectedness of $X$: Since $X$ contains a dense connected subset $X'$, then $X$ itself is connected. 

*Hausdorfness of $X$: (a) Suppose first that $x_1, x_2\in X$ project to distinct points in  $Y$. Then the existence of separating neighborhoods of $x_1, x_2$ follows from Hausdorfness of $Y$. (b) If $x_1, x_2$ project to the same point $y$ of $Y'$ then  use the fact that $y$ has a neighborhood $U\subset Y'$ with preimage a disjoint union of open subsets on each of which $p$ is injective. (c) Lastly, if $y$ belongs to $S$, then (in view of the argument in (b)) the only interesting case is when $x_1, x_2$ belong to the closure of the same punctured disk in $X$ which maps onto a punctured neighborhood of $y$ in $Y$. But this case is clear as well since the disk in ${\mathbb C}$ is Hausdorff.   
All this is just elementary general topology which Donaldson expects one to have mastered before reading his book. 


*The uniqueness is a bit less obvious but not hard either. The key is Riemann's extension theorem: An isolated singularity $z_0$ of a meromorphic function is removable if the function is bounded in a punctured neighborhood of $z_0$. 


Consider two "completions" of $X$: 
$$
\begin{array}{ccccc} 
& & X' &  &  \\
 & \swarrow &   & \searrow & \\
X_1 &  &   &   & X_2 \end{array}$$
such that $p$ extends to proper maps $p_k: X_k\to Y, k=1,2$. 
$$
\begin{array}{ccccc} 
X_1 & \leftarrow & X' & \rightarrow & X_2 \\
\downarrow{p_1} & & \downarrow{p} & & \downarrow{p_2}\\
Y & \leftarrow & Y' & \rightarrow & Y \end{array}
$$
We need to construct a biholomorphic map $X_1\to X_2$ forming a commutative diagram: 
$$
\begin{array}{ccccc} 
& & X' &  &  \\
 & \swarrow{i_1} &   & \searrow{i_2} & \\
X_1 &  & \longrightarrow  &   & X_2 \end{array}$$
We already have a biholomorphic map between dense subsets 
$$
X_1\supset i_1(X')\stackrel{f}{\to} i_2(X')\subset X_2
$$ 
The subsets $R_k:= X_k- i_k(X')$ are discrete in $X_k$, $k=1,2$. Now, for each $x_1\in R_1$ there is a neighborhood $V_1$ of $x_1$ in $X_1$ which is disjoint from  the rest of $R_1$. The image $W_2:=f(V_1-\{x_1\})\subset X_2$, by properness of $p_2$, forms a punctured neighborhood of a point $x_2\in R_2$. By shrinking $V_1$ if necessary, we can assume that $W_2$ lies in a coordinate neighborhood of $x_2$ (I just need to ensure that this larger neighborhood of $x_2$ in $X_2$ is biholomorphic to the unit disk). Thus, by Riemann's extension theorem, $f|V_1-\{x_1\}$ extends to a holomorphic map $V_1\to W_2\cup \{x_2\}$. Do this for every point in $R_1$. I will leave it to you to check that the resulting holomorphic extension $F: X_1\to X_2$ of $f$ is biholomorphic. 
A similar argument also proves the following theorem:
Definition. Let $X$ be a Riemann surface. A conformal completion of $X$ is a holomorphic embedding $\iota: X\to \hat{X}$ to another Riemann surface, such that:
a. $\hat{X}- \iota(X)$ is a discrete subset $S\subset \hat X$. 
b. For every properly embedded closed punctured disk $D\subset X$, the image $\iota(D)$ is a closed punctured neighborhood of some $s\in S$. 
Theorem. Every Riemann surface admits a conformal completion. Such a completion is unique in the natural sense. Moreover, every holomorphic automorphism $X\to X$ extends to a holomorphic automorphism of the completion.    
