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Recently we have demonstrated that high-precision polarization observations can detect the polarization resulting from the rotational distortion of a rapidly rotating B-type star. Here we investigate the extension of this approach to an A-type star. Linear-polarization observations of $α$ Oph (A5IV) have been obtained over wavelengths from 400 to 750 nm. They show the wavelength dependence expected for a rapidly-rotating star combined with a contribution from interstellar polarization. We model the observations by fitting rotating-star polarization models and adding additional constraints including a measured $v_e \sin{i}$. However, we cannot fully separate the effects of rotation rate and inclination, leaving a range of possible solutions. We determine a rotation rate $ω= Ω/Ω_ c$ between 0.83 and 0.98 and an axial inclination i > 60 deg. The rotation-axis position angle is found to be 142 $\pm$ 4 deg, differing by 16 deg from a value obtained by interferometry. This might be due to precession of the rotation axis due to interaction with the binary companion. Other parameters resulting from the analysis include a polar temperature Tp = 8725 $\pm$ 175 K, polar gravity $\log{g_p} = 3.93 \pm 0.08$ (dex cgs), and polar radius $R_{\rm p} = 2.52 \pm 0.06$ Rsun. Comparison with rotating-star evolutionary models indicates that $α$ Oph is in the later half of its main-sequence evolution and must have had an initial $ω$ of 0.8 or greater. The interstellar polarization has a maximum value at a wavelength ($λ_{\rm max}$) of $440 \pm 110$ nm, consistent with values found for other nearby stars.