We have measured the transverse asymmetry A{sub T'} in the quasi-elastic {sup 3}/rvec He/(/rvec e/,e') process with high precision at Q{sup 2}-values from 0.1 to 0.6 (GeV/c){sup 2}. The neutron magnetic form factor G{sub M}{sup n} was extracted at Q{sup 2}-values of 0.1 and 0.2 (GeV/c){sup 2} using a non-relativistic Faddeev calculation which includes both final-state interactions (FSI) and meson-exchange currents (MEC). Theoretical uncertainties due to the FSI and MEC effects were constrained with a precision measurement of the spin-dependent asymmetry in the threshold region of {sup 3}/rvec He/(/rvec e/,e'). We also extracted the neutron magnetic form factor G{sub M}{sup n} at Q{sup 2}-values of 0.3 to 0.6 (GeV/c){sup 2} based on Plane Wave Impulse Approximation calculations.