  In this paper, we study the spectrum of the complex Hill operator
$L=\frac{d^2}{dx^2}+q(x;\tau)$ in $L^2(\mathbb{R},\mathbb{C})$ with the
Darboux-Treibich-Verdier potential
\[q(x;\tau):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left(
x+z_0+\tfrac{\omega_{k}}{2};\tau \right),\] where $n_k\in\mathbb{Z}_{\geq 0}$
with $\max n_k\geq 1$ and $z_0\in\mathbb{C}$ is chosen such that $q(x;\tau)$
has no singularities on $\mathbb{R}$. For any fixed $\tau\in i\mathbb{R}_{>0}$,
we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to
guarantee that the spectrum $\sigma(L)$ is
  \[\sigma(L)=(-\infty, E_{2g}]\cup[E_{2g-1}, E_{2g-2}]\cup \cdots \cup[E_{1},
E_{0}],\quad E_j\in \mathbb{R},\]
  and hence generalizes Ince's remarkable result in 1940 for the Lam\'{e}
potential to the Darboux-Treibich-Verdier potential. We also determine the
number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1}$,
$E_{2j-2})$, which generalizes the recent result in \cite{HHV} where the
Lam\'{e} case $n_1=n_2=n_3=0$ was studied.
