Supermassive black holes in giant elliptical galaxies are remarkably faint given their expected accretion rates. This motivates models of radiatively inefficient accretion, due to either ion-electron thermal decoupling, generation of outflows that inhibit accretion, or settling of gas to a gravitationally unstable disk that forms stars in preference to feeding the black hole. The latter model predicts the presence of cold molecular gas in a thin disk around the black hole. Here we report Submillimeter Array observations of the nucleus of the giant elliptical galaxy M87 that probe 230 GHz continuum and CO(J=2--1) line emission. Continuum emission is detected from the nucleus and several knots in the jet, including one that has been undergoing flaring behavior. We estimate a conservative upper limit on the mass of molecular gas within ~100pc and +-400km/s line of sight velocity of the central black hole of ~8x10^6Msun, which includes an allowance for possible systematic errors associated with subtraction of the continuum. Ignoring such errors, we have a 3 sigma sensitivity to about 3x10^6Msun. In fact, the continuum-subtracted spectrum shows weak emission features extending up to 4 sigma above the RMS dispersion of the line-free channels. These may be artifacts of the continuum subtraction process. Alternatively, if they are interpreted as CO emission, then the implied molecular gas mass is ~5x10^6Msun spread out over a velocity range of 700km/s. These constraints on molecular gas mass are close to the predictions of the model of self-gravitating, star-forming accretion disks fed by Bondi accretion (Tan & Blackman 2005).