Cord blood harvests (for either stem cell transplants or stem cell research) don't involve retaining the cord, but the blood that's expressed from the cord.

As for keeping the umbilical cord, is it the entire umbilical cord? Or a segment of the cord? Or the "stump", the bit between the baby's belly button and the clamp that's applied at delivery. If it's the last then you'll leave the hospital with it still attached to your baby, and once it dries up and detaches (several days later) you can put it in the box.

If you need more of the cord then you could ask your OB to get it for you at the time of delivery if he/she doesn't see any reason for it to be sent to pathology for examination (for infection, vascular abnormalities, etc.). The placenta and attached cord are typically put in the trash in any case.