We report on 230 GHz (1.3 mm) very long baseline interferometry (VLBI) observations of M87 with the Event Horizon Telescope using antennas on Mauna Kea in Hawaii, Mt. Graham in Arizona, and Cedar Flat in California. For the first time, we have acquired 230 GHz VLBI interferometric phase information on M87 through measurement of the closure phase on the triangle of long baselines. Most of the measured closure phases are consistent with 0° as expected by physically motivated models for 230 GHz structure such as jet models and accretion disk models. The brightness temperature of the event-horizon-scale structure is ∼1 × 10¹⁰ K derived from the compact flux density of ∼1 Jy and the angular size of ∼40 μas ∼ 5.5 Rs, which is broadly consistent with the peak brightness of the radio cores at 1-86 GHz located within ∼10² Rs. Our observations occurred in the middle of an enhancement in very-high-energy (VHE) γ-ray flux, presumably originating in the vicinity of the central black hole. Our measurements, combined with results of multi-wavelength observations, favor a scenario in which the VHE region has an extended size of ∼20-60 Rs.