  In the weak field approximation, the nonsymmetric gravitational theory (NGT)
has, in addition to the Newtonian gravitational potential, a Yukawa-like
potential produced by the exchange of a spin $1^+$ boson between fermions. If
the range $r_0=\mu^{-1}$ is $25$ kpc, then this additional potential due to the
interaction with matter in the halos of galaxies can explain the flat rotation
curves of galaxies and the Tully-Fisher law ($L\sim v^4$) without the dark
matter hypothesis. Possible fits to clusters of galaxies and gravitational
lensing observations are discussed. The results are based on a linear
approximation to a new perturbatively consistent version of the NGT field
equations, which does not violate the weak equivalence principle.
