Hold your new string and old string together in an X shape, with the new string on top.

Fold the new string underneath the old string.

Fold the new string upwards and hold both ends of the string. Now it should be easy for you to see how to tie the ends into a knot. You should actually tie 2-3 knots, to make sure it won't come untied.

Then trim the excess string, and continue threading beads!