You ask a deceptively complex question. There are two copyright rights in music. The first is the composer’s rights. The second is the performer’s rights.
I will deal with the composer’s rights first. I looked up your song on Google and it appears the song was written by Edvard Grieg and first published in 1876. The copyright right has to have expired a long time ago. So you are free to use that piece of music.
Had the piece still been protected by copyright, you would have needed to acquire a license to use the piece. To record your own version of the piece, you would need a mechanical license. You would have to locate the owner of the copyright in the music. This can be hard to do. Fortunately the Harry Fox Agency (www.harrryfox.com) has taken it upon itself to try to identify the owner of every piece of music. No one can have a complete list, but Harry Fox comes very close. They do this as a promotional tool for themselves. Start with them. If you just wanted to make your own recording of the song, then the copyright law has a default license, called a compulsory license. You are free to negotiate your own license terms. But you can also insist on the terms and prices of the compulsory license.
If you wanted to perform the piece, say as part of a music concert, then there is a different license involved. Most music publishers in the U.S. have joined one of three societies, ASCAP, BMI or SESAC. Most performance venues obtain blanket license which cover anyone who performs there.
But if you want to include the music in a dramatic performance then you need a separate license. If you use the piece in a video, then you are using the piece in a dramatic performance and you would need to get permission if the piece were still protected by copyright.
Next is the performers’ license. If you are going to make your own recording, then you are fine because the copyright rights have expired.
But if you want to use an existing recording of the piece, then you also need a license from the performer, or more typically from the company that produced the recording. (Usually the artist has assigned the copyright rights to the producing company.) The copyright in any recording of the piece has probably not expired. If you intend to play the music as background music in a venue, then usually the ASCAP, BMI or SESAC license will work. Congress recently added an exemption for certain minor uses of the music.
If you want to include a piece and/or a recording of that piece in a dramatic performance then you need a separate license. I also often see entities using a music in a dramatic performance without getting permission because they mistakenly believe that the performance license includes dramatic performances which it does not.
As a practical matter, the owners of the composer and performance copyright rights in music rarely go after youtube videos, or local dramatic performances. But they could. And it depends on the music. You are more likely to get in trouble if you try to use a song written by and/or recorded by the Beatles or Bob Dylan, then say some struggling artist who needs the publicity.
The people who run the contest are wise to require you to show evidence of having obtained permission. They could get in trouble if you place a piece of music on their website without permission. (Although again, there are exemptions there too in certain cases.)
I tell my clients, when in doubt, get permission. Also keep in mind that with limited exceptions (like the compulsory license described above) the copyright owner is under no obligation to grant you permission or even to respond to your request. You can not use a piece that is protected by copyright until you have succeeded in obtaining permission, regardless of how hard you have tried to find the copyright owner and obtain permission.
This is a difficult area to write about. If I have misspoken or left something out, I encourage other attorneys to address the shortcoming.
For your video, you should write the Harry Fox Agency and ask for a license if the song is not public domain. If it is, and apparently Attorney Marshall has found it to be so, you do not need a license. Copy these answers, use the song, and attach the answers to the form so zooppa.com can see that you no license is needed to the song.
So, you state the song is public domain, attach the answers to the affidavit and submit your song. Simple, really.
So far, this is free to you. Until you pay a fee, I am not your lawyer and you are not my client, so you take any free advice at your sole risk. I am licensed in IL, MO, TX and am a Reg. Pat. Atty. so advice in any other jurisdiction is general advice and should be confirmed with an attorney licensed in that jurisdiction.
You need a license to the master recording rights, composition rights and possibly sync rights. You can find out who owns these rights by searching ASCAP, Harry Fox and BMI indexes and contact the publishing companies directly, or hire an intermediary to obtain the rights for you. Having done this before it can cost you $5,000 - $25,000 per song for a limited period of time, 3-6 months-ish, and that's if all the rights holders will license the song to you.
Whatever you decide to do. Don't use it first and ask for permission second as you may have to pay a penalty, or receive a cease and desist letter if they're going to deny you.
Gary has done a great job covering the copyright issues here. I would like to focus on the promotions law issues for brands. This contestant is asking good questions and attempting to follow the directives the "got milk" campaign established for submissions. Often, however, entrants are not as vigilant. It is essential that brands establish best practices for accepting user generated content in a contest. They need to be sure their rules adequately protect against liability should the contestant falsely believe something is in the public domain or submit a false license. In addition, they need to do their own internal vetting before making use of an entry to advertise or market their brand. This includes posting entries on a Facebook page or website even if not selected as a winning entry. The steps to take are complex and require the input of a specialized promotions lawyer who is conversant with the laws of the 50 states on sweepstakes and contests as well as intellectual property law issues such as have been raised here. This is not the proper forum to provide any detailed legal advice, and my answer here is just to get brands thinking about the legal issues involved with accepting user generated content.