  We present Pieri rules for the Jack polynomials in superspace. The
coefficients in the Pieri rules are, except for an extra determinant, products
of quotients of linear factors in $\alpha$ (expressed, as in the usual Jack
polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We
show that, surprisingly, the extra determinant is related to the partition
function of the 6-vertex model. We give, as a conjecture, the Pieri rules for
the Macdonald polynomials in superspace.
