  Recently proposed Born-Infeld (BI) theories of gravity assume a constant BI
parameter ($\kappa$). However, no clear consensus exists on the sign and value
of $\kappa$. Recalling the Brans-Dicke (BD) approach, where a scalar field was
used to generate the gravitational constant $G$, we suggest an extension of
Born-Infeld gravity with a similar Brans-Dicke flavor. Thus, a new action, with
$\kappa$ elevated to a spacetime dependent real scalar field, is proposed. We
illustrate this new theory in a cosmological setting with pressureless dust and
radiation as matter. Assuming a functional form of $\kappa(t)$, we numerically
obtain the scale factor evolution and other details of the background
cosmology. It is known that BI gravity differs from general relativity (GR) in
the strong-field regime but reduces to GR for intermediate and weak fields. Our
studies in cosmology demonstrate how, with this new, scalar-tensor BI gravity,
deviations from GR as well as usual BI gravity, may arise in the weak-field
regime too. For example, we note a late-time acceleration without any dark
energy contribution. Apart from such qualitative differences, we note that
fixing the sign and value of $\kappa$ is no longer a necessity in this theory,
though the origin of the BD scalar does remain an open question.
