  Let $N$ be a positive square-free integer such that the discrete group
$\Gamma_{0}(N)^{+}$ has genus one. In a previous article, we constructed
canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field
associated to $\Gamma_{0}(N)^{+}$ as well as an algebraic equation
$P_{N}(x_{N},y_{N}) = 0$ with integer coefficients satisfied by these
generators. In the present paper, we study the singular moduli problem
corresponding to $x_{N}$ and $y_{N}$, by which we mean the arithmetic nature of
the numbers $x_{N}(\tau)$ and $y_{N}(\tau)$ for any CM point $\tau$ in the
upper half plane $\mathbb{H}$. If $\tau$ is any CM point which is not
equivalent to an elliptic point of $\Gamma_{0}(N)^{+}$, we prove that the
complex numbers $x_{N}(\tau)$ and $y_{N}(\tau)$ are algebraic integers. Going
further, we characterize the algebraic nature of $x_{N}(\tau)$ as the generator
of a certain ring class field of $\mathbb{Q}(\tau)$ of prescribed order and
discriminant depending on properties of $\tau$ and level $N$. The theoretical
considerations are supplemented by computational examples. As a result, several
explicit evaluations are given for various $N$ and $\tau$, and further
arithmetic consequences of our analysis are presented. In one example, we
explicitly construct a set of minimal polynomials for the Hilbert class field
of $\mathbb{Q}(\sqrt{-74})$ whose coefficients are less than $2.2\times
10^{4}$, whereas the minimal polynomials obtained from the Hauptmodul of
$\textrm{PSL}(2,\mathbb{Z})$ has coefficients as large as $6.6\times 10^{73}$.
