This long article from Wired in 1996, Mother Earth Mother Board [1], would answer your question. It is long, it took me more than a couple of hours to read, but it's also my all-time favourite article from any magazine, and whenever the topic of submarine cables comes up on Hacker News I often see this link.
I read the entire article. It does not go into detail about the deepest parts of the ocean. The closest it gets is when it talked about the calculations required to lay cable from the UK to Africa, but this is still close to shore (I remember seeing a diagram showing how the cable angled down a steep slope into deep ocean, but this archived version doesn't seem to have pictures).
Edit: this is what it actually gives you as an answer in that article:
"The answer has to do with slack control. And most of what is known about slack control is known by Cable & Wireless Marine. AT&T presumably knows about slack control too, but Cable & Wireless Marine has twice as many ships and dominates the deep-sea cable-laying industry. The Japanese can lay cable in shallow water and can repair it anywhere. But the reality is that when you want to slam a few thousand kilometers of state-of-the-art optical fiber across a major ocean, you call Cable & Wireless Marine, based in England. That is pretty much what FLAG did several years ago."
"Beyond 1500 meters from the shoreline, builders typically lay unarmored cable directly on the seabed ... a few reach depths greater than 8000 meters."
Not much of an answer. How the heck do they lay it and control slack at greater than 8000 meters? How do they verify the cable is on the seabed at those depths?
In some cases the cable is buried on the sea floor by a special plow. It's incredible that the cable can be buried up to 6m deep in the mud on the bottom. Cable burial is intended to keep fishing equipment from snagging cables laid on the bottom.
Jets of water are used to bury the cable closer to shore, and if I recall correctly, Google used Kevlar on their latest cables to reinforce them against sharks.
