  We propose a new solvable one-dimensional complex PT-symmetric potential as
$V(x)= ig~ \mbox{sgn}(x)~ |1-\exp(2|x|/a)|$ and study the spectrum of
$H=-d^2/dx^2+V(x)$. For smaller values of $a,g <1$, there is a finite number of
real discrete eigenvalues. As $a$ and $g$ increase, there exist exceptional
points (EPs), $g_n$ (for fixed values of $a$) causing a scarcity of real
discrete eigenvalues, but there exists at least one. We also show these real
discrete eigenvalues as poles of reflection coefficient. We find that the
energy-eigenstates $\psi_n(x)$ satisfy (1): PT$\psi_n(x)=1 \psi_n(x)$ and (2):
PT$\psi_{E_n}(x)=\psi_{E^*_n}(x)$, for real and complex energy eigenvalues,
respectively.
