We confirmed the Josephson effect in vertically proximity-coupled junctions prepared by sandwiching a 43-nm-thick exfoliated graphite layer between two superconducting electrodes. Josephson coupling with well-controlled contact characteristics was established by thermal deposition of electrodes on both sides of freshly cleaved graphite surfaces. The genuine Josephson coupling through the c-axis graphite was confirmed by the critical current modulation in in-plane magnetic fields (Fraunhofer pattern) and the response to the microwave irradiation (Shapiro steps). This scheme can be potentially utilized to fabricate atomically thin vertically coupled nanodevices based on assorted cleavable layered materials.