Abstract The barred spiral galaxy NGC 613 has a star-forming ring in the center, and near-infrared observations have previously shown that the star formation activity on the eastern and western sides of the ring is asymmetric. We examined the dynamics and physical state of the molecular gas in the ring using high-resolution (∼15 pc) 12CO(1–0), 12CO(3–2), and 13CO(1–0) observations with ALMA. Using a dendrogram, we identified 111 molecular clouds in the bar and ring, and found that the virial parameter (αvir) gradually decreases (αvir < 2) toward the confluence of the northern bar and eastern ring, while the virial parameter is large (αvir > 2) around the corresponding confluence in the western side of the ring. A non-LTE analysis using RADEX showed that the temperature and density of the molecular gas increase downstream of the eastern point of confluence, whereas they change irregularly on the western side. The star formation efficiency is (1.7 ± 0.2) × 10−8 yr−1 in the eastern side of the ring, which is substantially higher than the (0.9 ± 0.3) × 10−8 yr−1 for the western side of the ring. Position–velocity diagrams show that the relative velocity of the gas between the bar and the ring is ∼70 km s−1 in the east, while it reaches ∼170 km s−1 in the west. We suggest that the star formation activity in the central region of NGC 613 depends strongly on the relative velocity of the gas between the bar and the ring.