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You may reasonably imagine that this is easy and computers have it all programmed in, but it's an illusion. Computer code for networking has developed in an organic fashion. At the end of 2008, the folks who manage Google's servers chose to skip the leap second in the proper sense, and instead they inserted 1000 one-millisecond delays (equivalent) during the course of the day on December 31, 2008. As their official blog described it:
"The solution we came up with came to be known as the 'leap smear.' We modified our internal NTP servers to gradually add a couple of milliseconds to every update, varying over a time window before the moment when the leap second actually happens. This meant that when it became time to add an extra second at midnight, our clocks had already taken this into account, by skewing the time over the course of the day. All of our servers were then able to continue as normal with the new year, blissfully unaware that a leap second had just occurred. We plan to use this 'leap smear' technique again in the future, when new leap seconds are announced by the IERS."

-FER

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