The transverse momentum cross section of $e^+e^-$ pairs in the $Z$-boson mass region of 66-116 GeV/$c^2$ is precisely measured using Run II data corresponding to 2.1 fb$^{-1}$ of integrated luminosity recorded by the Collider Detector at Fermilab. The cross section is compared with quantum chromodynamic calculations. One is a fixed-order perturbative calculation at ${\cal O}(\alpha_s^2)$, and the other combines perturbative predictions at high transverse momentum with the gluon resummation formalism at low transverse momentum. Comparisons of the measurement with calculations show reasonable agreement. The measurement is of sufficient precision to allow refinements in the understanding of the transverse momentum distribution.
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