This is a known problem with continuous loop string, especially on warbows
I'll share my findings.
1. Have plenty of tension on the string and serving material while serving.
2. When pulling the two ends of the loop together (where it makes a Y) start the serving with some figure 8 holding the two together, and overlap slightly over the loop serving. That works quite well, but can still pull apart with V high tension warbows.
3. What I've been doing recently is to bind the two sides together with some very fine linen thread (just 3 or 4 turns) and the touch it with low visc' super glue. Then serve over that. This seems good and reliable, the superglue doesn't adversely effect the serving or the string as it is both manmade and waxed ( I wouldn't recommend it with a natural string)... but it does soak into the linen forming a solid ring which clinches the two sides together.
Del
PS.. .don't tell the other... it's a secret