There are no laws specifically prohibiting the US Government from using mercenaries. There have been SCotUS interpretations that have led decisions to indicate that it is against the law, but there are no laws prohibiting it.
The use of mercenaries may lead to difficulties when the Geneva Convention is involved however and depending on the circumstances mercenaries may lose some or all of their convention rights if they are judged to be an Unlawful Combatant, but the sponsoring country is not penalized.
The Neutrality Act of 1939 and the Anti-Pinkerton Act of 1893 have both been cited by the SCotUS as indicating that the US should not use mercs, but the particulars of the case did not apply to the Federal Government.
In 1989 the UN passed the The International Convention against the Recruitment, Use, Financing and Training of Mercenaries but the US refuses to sign it.
So while mercenaries should be against the law is a valid position, the reality is that they are not against the law.
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abusus non tollit usum - A right should NOT be withheld from people on the basis that some tend to abuse that right.
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