  Starting from an ultracold sample of ground-state $^{23}$Na$^{87}$Rb
molecules, we investigate the lowest ro-vibrational level of the $b^3\Pi$ state
with high resolution laser spectroscopy. This electronic spin-forbidden
$X^1\Sigma^+ \leftrightarrow b^3\Pi$ transition features a nearly diagonal
Franck-Condon factor and has been proposed useful for probing and manipulating
the ultracold molecular gas. We measure the transition strength directly by
probing the ac Stark shift induced by near resonance light and determine the
total excited-state spontaneous emission rate by observing the loss of
molecules. From the extracted branching ratio and the theoretical modeling, we
find that the leakage to the continuum of the $a^3\Sigma^+$ state plays the
dominant role in the total transition linewidth. Based on these results, we
show that it is feasible to create optical trapping potentials for maximizing
the rotational coherence with laser light tuned to near this transition.
