Solution 1: Perhaps there's a ship pulling a cable attached to a reel, and it periodically drops the cable in the ocean with an anchor weight. The first few times they did this, the engineers miscalculated the length of the cable. As the ship sailed on, the reel got ripped from the ship's hull, the boat sank and the captain didn't have a connection to send an SOS. Ironically, the connection was at the bottom of the ocean, still attached to the reel.

Solution 2: Organize dolphins to pull cables across the Atlantic. Unfortunately, the spinner dolphin species was chosen, resulting in many miles of twisted fiber-optic cable. As you know, having kinks in fiber-optic cables is a bad thing.

Solution 3: Use the Blues Brothers' car (with a cop engine, cop suspension and cop brakes), roll up the windows and drive across the sea floor, laying the cable down. I think this is how they actually did it