The ends of the cable are on board the cable ship when this is done. For broken cables, cable ships find the ends, catch them with drags, haul the end onboard a ship, and splice in an additional section.

Animated video from a company that does this: [1]

[1] https://www.youtube.com/watch?v=QiwidMEv8CM

> How does the fusion splice process take place deep under water?

https://en.wikipedia.org/wiki/USS_Jimmy_Carter