Actually if you re-record it I don't think you need any permission. You just pay them a royalty rate. You'll save an arm and a leg by not using the original.
Mechanical Rights
Mechanical Rights, which are aquired from music publishers and copyright holders, give you the right to record, manufacture and distribute another copyright holder's musical work. Songwriters and publishers also receive payments from mechanical licenses. A mechanical license is written permission from the publisher to manufacture and distribute a record, CD or audio tape for a specific copyrighted composition. The amount of the royalty paid to a songwriter from a mechanical license is determined by how many recordings are sold.
Mechanical rights, although similar, are not to be confused with "master rights" or with "performance rights". "Master Rights" are granted by a record company in order to use an existing recording, while "performance rights" are granted by publishers or societies for the public performance of a song. Depending on the actual use of the copyright, you may need to obtain these additional rights, as well as the mechanical license.
What are Master Use Rights?
Master use rights are required for previously recorded material that you do not own or control. They can only be obtained from the owner of the master recording, usually a record company. It is recommended that you obtain the master use license from the owner prior to requesting a mechanical license.
Do I Need a Mechanical License?
If you are manufacturing and distributing copies of a song which you did not write, and you have not already reached an agreement with the song's publisher, you need to obtain a mechanical license. This is required under U.S. Copyright Law, regardless of whether or not you are selling the copies that you made.
Definitions
Audio/Visual Work: An industry term for film, television or any other audio/visual production such as presentations, Flash, Quicktime & other internet visual formats, videos, CDs, DVDs, etc.
Synchronization Rights: The right to use the music in timed relations with other visual elements in a film, video, television show/commercial, or other audio/visual production. In other words, the right to use the music as a soundtrack with visual images. Synchronization licenses are obtained from the publisher (or composer if no publisher) or the music library.
Master Use Rights: When you hear music on the radio or TV, this recording is known in the music industry as the "master recording". This is what is produced after all the musicians have played their parts and these parts have been "mixed" together for release. The recording of the master is also protected by copyright. A record label or music library owns this copyright, and can grant the right to use the recording in a compilation album, film soundtrack or other Audio/Visual medium. It grants the right to use the sound recording.
Performing Rights: Public Performance Right is the exclusive right the U.S. Copyright Law gives to the creator of a musical work or other copyrighted material authorizing the use in public. Every time a song is performed on a broadcast, there is a public performance. This public performance is licensed by performing rights organizations (BMI/ASCAP/etc.) or directly from the copyright holder as a direct license.
Mechanical Rights: License granting the right to record and release a specific composition at an agreed-upon fee per unit manufactured and sold. Right to use a song owned by someone else on a recording.
Grand Rights: Term used to describe "dramatic" performing rights. This would include musical comedies (Broadway and off-Broadway), operas, operettas, ballets, as well as renditions of musical compositions in a dramatic setting where there is narration, a plot and/or costumes and scenery. The copyright owner has the exclusive right to issue licenses and collect fees for grand rights. Performance Rights Organizations do not collect performing rights royalties for this use, and are licensed directly from either the composer or publisher.
Direct License: A license obtained directly from the copyright owner or publisher where the Performing Rights are paid directly to the copyright owner by the Licensee. With a Direct License, no royalties are collected by, or paid to, the Performing Rights Organizations (BMI/ASCAP/etc.).
Copyright: The exclusive right, granted by law for a stated period, usually until 70 years after the death of the surviving author of the work, to make, dispose of, and otherwise control copies of literary, musical, dramatic, pictorial and other copyrightable works.