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In this paper we determine the topology of the moduli space$\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannianmetric 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 anatural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for acertain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which isnon-normal for $m>1$.

Moduli of spherical tori with one conical point