From the link... "Current plans call for a base tower approximately 50 km tall -- the cable would be tethered to the top. To keep the cable structure from tumbling to Earth, it would be attached to a large counterbalance mass beyond geostationary orbit, perhaps an asteroid moved into place for that purpose. "
See? All we have to do now is figure out how to capture an asteroid, move it TOWARD EARTH to within 25,000 miles, and stablize it into earth orbit. And tie a rope to it.
That could take weeks.