Here's a great question, which can also be a difficult subject to follow. The term "Public Domain" makes it sound like anything in this category belongs to the public. But that's far from being true.
This is another one of those topics that has differences based on country location. So, you will have to dig deeper into your country's laws to get the best answer. Here's a starting point for those of you here in the United States.
Works published in the United States with a copyright date of 1922 or earlier are in the the public domain.
Do not be fooled by the term "Public Domain"... it can be tricky. Just because a musical composition or arrangement may be in the proverbial public domain, does not mean that you automatically can use it willy nilly in your production without paying for it. You must be absolutely certain that the specific recording you are using is also in the public domain.
Chances are, that the recording you have selected is not in the public domain. Permission and proper clearance from the owners of this recording must be obtained.
Of course you are legally free to spend your own money and have any public domain musical composition arranged, performed and recorded. Then, you would be the proud owner of this particular master recording and everyone else would need your permission to use it.
Every musical composition owned and provided by American Music Company is without exception copyrighted and is not in the public domain.