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The stabilizing effect of a sheared axial flow on the m=1 kink instability in Z pinches has been studied numerically with a linearized ideal magnetohydrodynamic model to reveal that a sheared axial flow stabilizes the kink mode when the shear exceeds a threshold. The sheared flow stabilizing effect is investigated with the ZaP (Z-Pinch) Flow Z-pinch experiment at the University of Washington. An axially flowing Z pinch is generated with a 1 m coaxial accelerator coupled to a pinch assembly chamber. The plasma assembles into a pinch 50 cm long with a radius of approximately 1 cm. An azimuthal array of surface mounted magnetic probes located at the midplane of the pinch measures the fluctuation levels of the azimuthal modes m=1, 2, and 3. After the pinch assembles a quiescent period is found where the mode activity is significantly reduced. Optical images from a fast framing camera and a ruby holographic interferometer indicate a stable, discrete pinch plasma during this time. Multichord Doppler shift measurements of impurity lines show a large, sheared flow during the quiescent period and low, uniform flow profiles during periods of high mode activity. Z-pinch plasmas have been produced that are globally stable for over 700 times the theoretically predicted growth time for the kink mode of a static Z pinch. The plasma has a sheared axial flow that exceeds the theoretical threshold for stability during the quiescent period and is lower than the threshold during periods of high mode activity.