This is a very interesting assumption
http://www.howstuffworks.com/tech-myths/5-myths-about-google...
They literally bring part of the cable into the sub and work on it from there.
Apparently the procedure is to position the USS Jimmy Carter over the cable, send out a remote vehicle to grab the cable and pull it up to the sub, splice in tapping equipment, and then drop the cable.
Intercontinental cables are less than one inch in diameter in deep water. If the sub plants the tap in deep water, it's extremely unlikely that anyone would ever discover it. There are a few cable repair ships that pull broken cables to the surface and fix them, but other than that... No one can reach it and no one will bother it.
The most interesting thing is actually getting the data back to the U.S. Do they run a separate cable alongside the existing cable? Getting SMALL bits of data back to the U.S. is easy, can just broadcast it. But LARGE amounts of data? Tricky.
Even if you pull the cable up to repair it, it is unlikely that you would discover the tap. The device used in Operation Ivy Bells was designed to detach from the cable if the cable was lifted. Non-intrustive tapping might not be possible with fiber (it isn't, as far as I know), but I expect they have other mechanisms to avoid discovery. Perhaps lifting the cable would cause the cable to "snap" on either side of the tap, before the device could be lifted.
