  This paper deals with the inverse spectral problem for a non-self-adjoint
Sturm-Liouville operator with discontinuous conditions inside the interval. We
obtain that if the potential $q$ is known a priori on a subinterval $ \left[
b,\pi \right] $ with $b\in \left( d,\pi \right] $ or $b=d$, then $h,$ $\beta ,$
$\gamma \ $and $q$ on $\left[ 0,\pi \right] \ $can be uniquely determined by
partial spectral data consisting of a sequence of eigenvalues and a subsequence
of the corresponding generalized normalizing constants or a subsequence of the
pairs of eigenvalues and the corresponding generalized ratios. For the case
$b\in \left( 0,d\right) ,$ a similar statement holds if $ \beta ,$ $\gamma \
$are also known a priori. Moreover, if $q$ satisfies a local smoothness
condition, we provide an alternative approach instead of using the high-energy
asymptotic expansion of the Weyl $m$-function to solve the problem of missing
eigenvalues and norming constants.
