  In this paper we determine the topology of the moduli space
$\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian
metric of constant curvature $1$ and one conical point of angle
$2\pi\vartheta$. In particular, for $\vartheta\in (2m-1,2m+1)$ non-odd,
$\mathcal{MS}_{1,1}(\vartheta)$ is connected, has orbifold Euler characteristic
$-m^2/12$, and its topology depends on the integer $m>0$ only. For
$\vartheta=2m+1$ odd, $\mathcal{MS}_{1,1}(2m+1)$ has $\lceil{m(m+1)/6}\rceil$
connected components. For $\vartheta=2m$ even, $\mathcal{MS}_{1,1}(2m)$ has a
natural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for a
certain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which is
non-normal for $m>1$.
