If your Ubuntu machine listens to NTP and synchronizes time with the Internet, the system will automatically adjust itself for the difference in time.
While your computer might not necessarily be aware of the fact that a leap second happened, it will catch and record the event as NTP servers push the change throughout the internet.
If you don't feel like the extra second is being added, you can force an update using the following command:
sudo ntpdate -s time.nist.gov
If you are not in the US, replace
time.nist.gov with the appropriate server as listed on this site. For the UK specifically, see this page for a list of valid servers.
Alternatively, if you're on a modern version of Ubuntu, there's a builtin utility called
timedatectl. By default, this runs once automatically on startup. Therefore, a quick reboot can force a sync if necessary.