It will be added after 23:59:59 UTC, the next second will be 23:59:60
It's not a big deal and computers normally know how to handle it, however in the past some poorly coded software (mainly java based) has experienced problems after seeing a leap second. If you are running such software please be advised to check if it still works correctly after the leap second is inserted, or if it needs a restart.
Tuesday, June 16, 2015