The European copyright laws grant ownership of a recorded track for fifty years, which Love Me Do just passed. That means that, starting January 1 of 2013, anyone who wants to put out the track is free to do so.
That law may be changed before the end of this year, moving the threshold to 70 years, which will help records released in 1963, but it is not expected that the new protection will be retroactive. The current U.S. copyright law grants 95 years of protection.
The first to take advantage of the new status is a small company called Digital Remasterings who have included Love Me Do on a release of early Beatles recordings including the often released Hamburg Star Club live show.
Another who has released the song is Pristine Classical who normally specializes in remastering and releasing public domain classical pieces but has broken that policy for the Beatles' song. According to many sources, they are doing it more in protest of the the pending updated limit which will reduce the number of recordings available for their library.