We report detection of strong infrared thermal emission from the nearby (d=19 pc) transiting extrasolar planet HD 189733b by measuring the flux decrement during its prominent secondary eclipse. A 6 hr photometric sequence using Spitzer's infrared spectrograph in peak-up imaging mode at 16 μm shows the secondary eclipse depth to be 0.551%+/-0.030%, with accuracy limited by instrumental baseline uncertainties, but with 32 σ precision (σ=0.017%) on the detection. The 16 μm brightness temperature of this planet (1117+/-42 K) is very similar to the Spitzer detections of TrES-1 and HD 209458b, but the observed planetary flux (660 μJy) is an order of magnitude greater. This large signal will allow a detailed characterization of this planet in the infrared. Our photometry has sufficient signal-to-noise ratio (~400 per point) to motivate a search for structure in the ingress/egress portions of the eclipse curve, caused by putative thermal structure on the disk of the planet. We show that by binning our 6 s sampling down to ~6 minute resolution, we detect the modulation in the intensity derivative during ingress/egress due to the overall shape of the planet, but our sensitivity is not yet sufficient to distinguish between realistic models of the temperature distribution across the planet's disk. We point out the potential for extending Spitzer secondary eclipse detections down to the regime of transiting hot Neptunes, if such systems are discovered among nearby lower main-sequence stars.