arXiv:1002.3723v2 [math-ph] 16 Jun 2010 1 New exact solution of the one dimensional Dirac Equation for the Woods-Saxon potential within the effective mass case $_{O. Panella}$, 1 $_{S. Biondini}$, 2 and A. Arda 2, 3 $^{1}$Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italy $^{2}$Dipartimento di Fisica, Universit` a degli Studi di Perugia, Via A. Pascoli, I-06123, Perugia, Italy $^{3}$Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey (Dated: October 26, 2018) We study the one-dimensional Dirac equation in the framework of a position dependent mass under the action of a Woods-Saxon external potential. We find that constraining appropriately the mass function it is possible to obtain a solution of the problem in terms of the hypergeometric function. The mass function for which this turns out to be possible is continuous. In particular we study the scattering problem and derive exact expressions for the reflection and transmission coefficients which are compared to those of the constant mass case. For the very same mass function the bound state problem is also solved, providing a transcendental equation for the energy eigenvalues which is solved numerically. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd Abstract