  Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to
be those in the boundary of (but not in) the variety of solutions to the
``modified'' qYB equation, the latter being analogous to the modified classical
Yang-Baxter (cYB) equation. We construct, for a large class of solutions $r$ to
the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary
solutions to the qYB equation of the form $I+tr+ t^2r_{2} + ...$. In the last
section we list and give quantizations for all classical r-matrices in $sl(3)
\wedge sl(3)$.
