  We construct new examples of Krall discrete orthogonal polynomials, i.e.,
orthogonal polynomials with respect to a measure which are also eigenfunctions
of a higher order difference operator. The new examples include the orthogonal
polynomials with respect to the measures obtained from the Meixner measure
$\rho_{a,c}$ and Hahn measure $\rho_{a,b,N}$ by removing a finite number of
their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$
of the Hahn measure are positive integers. From the new Krall-Meixner families
we construct new families of exceptional Meixner and Laguerre polynomials.
