The MiniBooNE experiment at Fermilab has amassed the largest sample to date of π0s produced in neutral current (NC) neutrino–nucleus interactions at low energy. This Letter reports a measurement of the momentum distribution of π0s produced in mineral oil (CH2) and the first observation of coherent π0 production below 2 GeV. In the forward direction, the yield of events observed above the expectation for resonant production is attributed primarily to coherent production off carbon, but may also include a small contribution from diffractive production on hydrogen. Integrated over the MiniBooNE neutrino flux, the sum of the NC coherent and diffractive modes is found to be (19.5±1.1(stat)±2.5(sys))% of all exclusive NC π0 production at MiniBooNE. These measurements are of immediate utility because they quantify an important background to MiniBooNE's search for νμ→νe oscillations.