  We examine several properties of the Berry curvature for the organic
conductor $\alpha$-(BEDT-TTF)$_2$I$_3$ consisting of four bands, which exhibits
a zero-gap state with Dirac cones. By adding a small potential acting on two
molecular sites, which breaks the inversion symmetry, it is shown that the
curvature for the Dirac particles displays a pair of peaks with opposite signs
and that each peak increases with decreasing potential. The Berry curvature
originating from the property of the wave function is analyzed using a reduced
Hamiltonian with a 2x2 matrix based on the Luttinger-Kohn representation, which
describes a pair of Dirac particles between the conduction band and the valence
band. Two types of velocity fields in the reduced Hamiltonian, whose vector
product gives the Berry curvature, rotate around the Dirac point as a vortex.
It is also shown that the other bands exhibit another pair of peaks of Dirac
particles with a tendency toward merging.
