The story basically has a billionaire wanting to build a space elevator. He begins digging a massive hole on earth and another group building the cable on an asteroid. He then propels the asteroid towards earth landing the cable in the hole and exploding charges to bury the end of the cable.
I think the story you're remembering is in the novel The Web Between Worlds by Charles Sheffield. Amazon summary:
Rob Merlin, a famous and sought-after engineer, is recruited by Darius Regulo, the aged, wealthy, and eccentric "king of space" to construct a bridge between earth and space.
Construction concludes with the end of the space tether plunging into a deep hole near Quito, Ecuador and explosives being detonated to cause landslides to refill the hole.