We observed supernova 1987A (SN 1987A) with the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope (HST) in 1999 September, and again with the Advanced Camera for Surveys (ACS) on the HST in 2003 November. No point source is observed in the remnant. We obtain a limiting flux of F_opt < 1.6 x 10^{-14} ergs/s/cm^2 in the wavelength range 2900-9650 Angstroms for any continuum emitter at the center of the supernova remnant (SNR). It is likely that the SNR contains opaque dust that absorbs UV and optical emission, resulting in an attenuation of ~35% due to dust absorption in the SNR. Taking into account dust absorption in the remnant, we find a limit of L_opt < 8 x 10^{33} ergs/s. We compare this upper bound with empirical evidence from point sources in other supernova remnants, and with theoretical models for possible compact sources. Bright young pulsars such as Kes 75 or the Crab pulsar are excluded by optical and X-ray limits on SN 1987A. Of the young pulsars known to be associated with SNRs, those with ages < 5000 years are all too bright in X-rays to be compatible with the limits on SN 1987A. Examining theoretical models for accretion onto a compact object, we find that spherical accretion onto a neutron star is firmly ruled out, and that spherical accretion onto a black hole is possible only if there is a larger amount of dust absorption in the remnant than predicted. In the case of thin-disk accretion, our flux limit requires a small disk, no larger than 10^{10} cm, with an accretion rate no more than 0.3 times the Eddington accretion rate. Possible ways to hide a surviving compact object include the removal of all surrounding material at early times by a photon-driven wind, a small accretion disk, or very high levels of dust absorption in the remnant.
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