  We propose and experimentally investigate a scheme for narrow-line cooling of
KRb molecules in the rovibrational ground state. We show that the
spin-forbidden $\mathrm{X^1\Sigma^+} \rightarrow \mathrm{b^3\Pi_{0^+}}$
transition of KRb is ideal for realizing narrow-line laser cooling of molecules
because it has highly diagonal Franck-Condon factors and narrow linewidth. In
order to confirm the prediction, we performed the optical and microwave
spectroscopy of ultracold $^{41}$K$^{87}$Rb molecules, and determined the
linewidth ($2\pi\times$ 4.9(4) kHz) and Franck-Condon factors for the
$\mathrm{X^1\Sigma^+} (v''=0) \rightarrow \mathrm{b^3\Pi_{0^+}} (v'=0)$
transition (0.9474(1)). This result opens the door towards all-optical
production of polar molecules at sub-microkelvin temperatures.
