  A minimal-length scenario can be considered as an effective description of
quantum gravity effects. In quantum mechanics the introduction of a minimal
length can be accomplished through a generalization of Heisenberg's uncertainty
principle. In this scenario, state eigenvectors of the position operator are no
longer physical states and the representation in momentum space or a
representation in a quasiposition space must be used. In this work, we solve
the Schroedinger equation with Dirac $\delta$-function potential in
quasiposition space. We calculate the bound state energy and the coefficients
of reflection and transmission for scattering states. We show that leading
corrections are of order of the minimal length $({\sl O}(\sqrt{\beta}))$ and
the coefficients of reflection and transmission are no longer the same for the
Dirac delta well and barrier as in ordinary quantum mechanics. Furthermore,
assuming that the equivalence of the 1s state energy of the hydrogen atom and
the bound state energy of the Dirac $\delta$-function potential in 1-dim is
kept in a minimal-length scenario, we also find that the leading correction
term for the ground state energy of the hydrogen atom is of order of the
minimal length and $\Delta x_{min} \le 10^{-25}$ m.
