The Beatles song "Love Me Do" is in the Public Domain. Which means it may be covered by anyone with the means to record, without approval or compensation for those involved in its creation*. The European Union’s current copyright law states that any song released after January 1st, 1963 won’t become public domain for 70 years. The Beatles’ debut single “Love Me Do” and its B-Side “P.S. I Love You” was released in October 1962, which means it falls out of copyright protection after 50 years.
Here's another article about the subject: www2.gibson.com/News-Lifestyle/News/en-us/Beatles-in-Public-Domain
*Copyrights and laws confuse me. I'm no lawyer. For proper legal advise please consult a qualified lawyer. If you plan to break the law be sure you don't get caught and your back-up plan should include a relative that is a judge.
Since I have you here reading about the Beatles, did you hear Nirvana and Paul McCartney recorded a song together? It's called "Cut Me Some Slack" and you can find out more about that tune here: kdkb.com/dick-havoc/blog/cut-me-some-slack