We report the first measurement of the parity-violating elastic electron scattering asymmetry on ^{27}Al. The ^{27}Al elastic asymmetry is A_{PV}=2.16±0.11(stat)±0.16(syst) ppm, and was measured at ⟨Q^{2}⟩=0.02357±0.00010 GeV^{2}, ⟨θ_{lab}⟩=7.61°±0.02°, and ⟨E_{lab}⟩=1.157 GeV with the Q_{weak} apparatus at Jefferson Lab. Predictions using a simple Born approximation as well as more sophisticated distorted-wave calculations are in good agreement with this result. From this asymmetry the ^{27}Al neutron radius R_{n}=2.89±0.12 fm was determined using a many-models correlation technique. The corresponding neutron skin thickness R_{n}-R_{p}=-0.04±0.12 fm is small, as expected for a light nucleus with a neutron excess of only 1. This result thus serves as a successful benchmark for electroweak determinations of neutron radii on heavier nuclei. A tree-level approach was used to extract the ^{27}Al weak radius R_{w}=3.00±0.15 fm, and the weak skin thickness R_{wk}-R_{ch}=-0.04±0.15 fm. The weak form factor at this Q^{2} is F_{wk}=0.39±0.04.