  We present a new method for determining the intrinsic natural linewidth or
lifetime of exceptionally long-lived optical excited states. Such transitions
are key to the performance of state-of-the-art atomic clocks, have potential
applications in searches for fundamental physics and gravitational wave
detectors, as well as novel quantum many-body phenomena. With longer lifetime
optical transitions, sensitivity is increased, but so far it has proved
challenging to determine the natural lifetime of many of these long lived
optical excited states because standard population decay detection techniques
become experimentally difficult. Here, we determine the ratio of a poorly known
ultranarrow linewidth transition ($^3$P$_0$ to $^1$S$_0$ in $^{87}$Sr) to that
of another narrow well known transition ($^3$P$_1$ to $^1$S$_0$) by coupling
the two transitions to a single optical cavity and performing interleaved
nondestructive measurements of the interaction strengths of the atoms with
cavity modes near each transition frequency. We use this approach to determine
the natural linewidth of the clock transition $^3$P$_0$ to $^1$S$_0$ in
$^{87}$Sr to be $\gamma_0/(2\pi) = 1.35(3)~$mHz or $\tau= 118(3)~$s. The
30$~\mu$Hz resolution implies that we could detect states with lifetimes just
below 2 hours, and with straightforward future improvements, we could detect
states with lifetimes up to 15 hours, using measurement trials that last only a
few hundred milliseconds, eliminating the need for long storage times in
optical potentials.
