  The electronic band structure of graphene in the presence of spin-orbit
coupling and transverse electric field is investigated from first principles
using the linearized augmented plane-wave method. The spin-orbit coupling opens
a gap at the $K(K')$-point of the magnitude of 24 $\mu$eV (0.28 K). This
intrinsic splitting comes 96% from the usually neglected $d$ and higher
orbitals. The electric field induces an additional (extrinsic)
Bychkov-Rashba-type splitting of 10 $\mu$eV (0.11 K) per V/nm, coming from the
$\sigma$-$\pi$ mixing. A 'mini-ripple' configuration with every other atom is
shifted out of the sheet by less than 1% differs little from the intrinsic
case.
