You don't clamp it. You wait (IIRC) 15-30 minutes before doing anything to it, to give it enough time to drain. The immediacy with which the cord is routinely clamped and cut rose quite a bit in the 80's and 90's. It only makes sense that you shouldn't tamper with nature so much as to cut off the cord from the baby immediately after birth, while there's still blood flowing through it.

I'm not really sure why doctors started doing that, but it reminds me of our society's pressure towards having everything natural seem perfect and sort of sterile, like when our grocery stores reject all but the most flawless produce. We wouldn't want our babies to be handed to us with all sorts of icky, biological stuff like still-intact umbilical cords attached.