  In an economical system with only two heavy right handed neutrinos, we
postulate a new texture for $3 \times 2$ Dirac mass matrix $m_D$. This model
implies one massless light neutrino and thus displays only two patterns of mass
spectrum for light neutrinos, namely hierarchical or inverse-hierarchical. Both
the cases can correctly reproduce all the current neutrino oscillation data
with a unique prediction $m_{\nu_e \nu_e} = \frac{\sqrt{\Delta
m^2_{solar}}}{3}$ and $\sqrt{\Delta m^2_{atm}}$ for the hierarchical and the
inverse-hierarchica cases, respectively, which can be tested in next generation
neutrino-less double beta decay experiments. Introducing a single physical CP
phase in $m_D$, we examine baryon asymmetry through leptogenesis.
Interestingly, through the CP phase there are correlations between the amount
of baryon asymmetry and neutrino oscillation parameters. We find that for a
fixed CP phase, the hierarchical case also succeeds in generating the observed
baryon asymmetry in our universe, plus a non-vanishing $U_{e3}$ which is
accessible in future baseline neutrino oscillation experiments.
