You'll probably need to tie the ends together. Clothes line is designed not to stick to much so you're unlikley to be able to glue it and even if you did you risk the glue coming off onto clothes. I'm no expert on the material but you could try a figure of eight or bowline, both a fairly tough.
If you don't like that you could try some sort of lashing or clip around the 2 ends. Failing that you could always replace the whole line.