  The astrophysical $S$-factor for the radiative capture $d(p,\gamma)^3$He in
the energy-range of interest for Big Bang Nucleosynthesis (BBN) is calculated
using an {\it ab-initio} approach. The nuclear Hamiltonian retains both two-
and three-nucleon interactions - the Argonne $v_{18}$ and the Urbana IX,
respectively. Both one- and many-body contributions to the nuclear current
operator are included. The former retain for the first time, besides the $1/m$
leading order contribution ($m$ is the nucleon mass), also the next-to-leading
order term, proportional to $1/m^3$. The many-body currents are constructed in
order to satisfy the current conservation relation with the adopted Hamiltonian
model. The hyperspherical harmonics technique is applied to solve the $A=3$
bound and scattering states. A particular attention is used in this second case
in order to obtain, in the energy range of BBN, an uncertainty on the
astrophysical $S$-factor of the order or below $\sim$1 %. Then, in this energy
range, the $S$-factor is found to be $\sim$10 % larger than the currently
adopted values.Part of this increase (1-3 %) is due to the $1/m^3$ one-body
operator, while the remaining is due to the new more accurate scattering wave
functions. We have studied the implication of this new determination for the
$d(p,\gamma)^3$He $S$-factor on deuterium primordial abundance. We find that
the predicted theoretical value for $^2$H/H is in excellent agreement with its
experimental determination, using the most recent determination of baryon
density of Planck experiment, and with a standard number of relativistic
degrees of freedom $N_{\rm eff}=3.046$ during primordial nucleosynthesis.
