$\newcommand{\ZZ}{\mathbb{Z}}\newcommand{\RR}{\mathbb{R}}$Let $S = \Sigma_2$ be the genus two surface.  In this case, $\ZZ^4$ is the deck group of the desired covering.  Consider $\ZZ^4$ inside of $\RR^4$ and add to these points the usual edges labelled $a, b, c, d$ parallel to the four coordinate axes.  This gives a Cayley graph for $\ZZ^4$. 

Next, starting at every vertex of the graph we attach a two-cell via the attaching map $abcdABCD$ (capital letters denote inverses).  This is possible because the boundary word describes a closed loop in the graph.  Let $S'$ be the resulting two-complex.  Every edge of $S'$ meets a pair of two-cells while every vertex meets eight two-cells.  The eight corners give the vertex a disk neighborhood in $S'$.

Thus $S'$ is a surface.  Taking the quotient by the action of $\ZZ^4$ gives the original surface $S$.  By the Galois correspondence, $S'$ is the desired covering space.  Note that $S'$ is quasi-isometric to $\ZZ^4$ so it is one-ended.  The loops $abAABa$ and $cdCCDc$, based at the origin, meet in exactly one point.  Thus $S'$ has genus, and so has infinite genus. 

This construction works in any genus.  When $g = 1$ the construction produces the universal cover.