WASP-12b is a transiting hot Jupiter on a 1.09-day orbit around a late-F star. Since the planets discovery in 2008, the time interval between transits has been decreasing by $29pm 2$ msec year$^{-1}$. This is a possible sign of orbital decay, although the previously available data left open the possibility that the planets orbit is slightly eccentric and is undergoing apsidal precession. Here, we present new transit and occultation observations that provide more decisive evidence for orbital decay, which is favored over apsidal precession by a $Deltamathrm{BIC}$ of 22.3 or Bayes factor of 70,000. We also present new radial-velocity data that rule out the R{o}mer effect as the cause of the period change. This makes WASP-12 the first planetary system for which we can be confident that the orbit is decaying. The decay timescale for the orbit is $P/dot{P} = 3.25pm 0.23$ Myr. Interpreting the decay as the result of tidal dissipation, the modified stellar tidal quality factor is $Q_star = 1.8 times10^{5}$.
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