Starting from an ultracold sample of ground-state $^{23}$Na$^{87}$Rb molecules, we investigate the lowest ro-vibrational level of the $b^3Pi$ state with high resolution laser spectroscopy. This electronic spin-forbidden $X^1Sigma^+ leftrightarrow b^3Pi$ 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 $a^3Sigma^+$ 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.
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