We present Atacama Large Millimeter Array CO(3$-$2) and HCO$^+$(4$-$3) observations covering the central $1rlap{.}5$$times$$1rlap{.}5$ region of the Orion Nebula Cluster (ONC). The unprecedented level of sensitivity ($sim$0.1 mJy beam$^{-1}$) and angular resolution ($sim$$0rlap{.}09 approx 35$ AU) of these line observations enable us to search for gas-disk detections towards the known positions of submillimeter-detected dust disks in this region. We detect 23 disks in gas: 17 in CO(3$-$2), 17 in HCO$^+$(4$-$3), and 11 in both lines. Depending on where the sources are located in the ONC, we see the line detections in emission, in absorption against the warm background, or in both emission and absorption. We spectrally resolve the gas with $0.5$ km s$^{-1}$ channels, and find that the kinematics of most sources are consistent with Keplerian rotation. We measure the distribution of gas-disk sizes and find typical radii of $sim$50-200 AU. As such, gas disks in the ONC are compact in comparison with the gas disks seen in low-density star-forming regions. Gas sizes are universally larger than the dust sizes. However, the gas and dust sizes are not strongly correlated. We find a positive correlation between gas size and distance from the massive star $theta^1$ Ori C, indicating that disks in the ONC are influenced by photoionization. Finally, we use the observed kinematics of the detected gas lines to model Keplerian rotation and infer the masses of the central pre-main-sequence stars. Our dynamically-derived stellar masses are not consistent with the spectroscopically-derived masses, and we discuss possible reasons for this discrepancy.
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