  We recently showed that a new gauge boson $\gamma_B$ coupling only to baryon
number is phenomenologically allowed, even if $m_B<m_Z$ and $\alpha_B \approx
0.2$. In our previous work we assumed that kinetic mixing between the baryon
number and hypercharge gauge bosons (via an $F^{\mu\nu}_B F_{\mu\nu}^Y$ term)
was small enough to evade constraints from precision electroweak measurements.
In this paper we propose a class of models in which this term is naturally
absent above the electroweak scale. We show that the generation of a mixing
term through radiative corrections in the low-energy effective theory does not
lead to conflict with precision electroweak measurements and may provide a
leptonic signal for models of this type at an upgraded Tevatron.
