  Big Bang Nucleosynthesis (BBN) is studied within the framework of a
two-parameter family of tensor-scalar theories of gravitation, with nonlinear
scalar-matter coupling function a(phi).
  We run a BBN code modified by tensor-scalar gravity, and impose that the
theoretically predicted BBN yields of Deuterium, Helium and Lithium lie within
some conservative observational ranges. It is found that large initial values
of a(phi) (corresponding to initial cosmological expansion rates much larger
than standard) are compatible with observed BBN yields. However, the
BBN-inferred upper bound on the cosmological baryon density is insignificantly
modified by considering tensor-scalar gravity. Taking into account the effect
of e^+ e^- annihilation together with the subsequent effect of the
matter-dominated era (which both tend to decouple phi from matter), we find
that the present value of the scalar coupling, i.e. the present level of
deviation from Einstein's theory, must be, for compatibility with BBN, smaller
than alpha_0^2 < 10^{-6.5} beta^{-1} (Omega_{matter} h^2 / 0.15)^{-3/2} when
beta > 0.5.
