We present Spitzer Space Telescope 3.6 and 4.5 micron observations of the binary neutron star merger GW170817 at 43, 74, and 264 days post-merger. Using the final observation as a template, we uncover a source at the position of GW170817 at 4.5 micron with a brightness of 22.9+/-0.3 AB mag at 43 days and 23.8+/-0.3 AB mag at 74 days (the uncertainty is dominated by systematics from the image subtraction); no obvious source is detected at 3.6 micron to a 3-sigma limit of >23.3 AB mag in both epochs. The measured brightness is dimmer by a factor of about 2-3 times compared to our previously published kilonova model, which is based on UV, optical, and near-IR data at <30 days. However, the observed fading rate and color (m_{3.6}-m_{4.5}> 0 AB mag) are consistent with our model. We suggest that the discrepancy is likely due to a transition to the nebular phase, or a reduced thermalization efficiency at such late time. Using the Spitzer data as a guide, we briefly discuss the prospects of observing future binary neutron star mergers with Spitzer (in LIGO/Virgo Observing Run 3) and the James Webb Space Telescope (in LIGO/Virgo Observing Run 4 and beyond).
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