First, both of the prior answers are wrong.
Second, being “based in the UK” is irrelevant. I’ll assume that you mean that the freelancer is resident in the UK, within the meaning of the US-UK income tax treaty. (He could also be a treaty resident of the US — or a resident of France, for that matter — conducting a business “based in the UK.” In either case, the result would be completely different.)
Third, I’ll assume that the “freelancer” is in fact not treated as an employee in either the UK or the US.
As a UK-resident self-employed individual under the treaty, the freelancer would be taxed in the UK on all of his income from that business. In addition, he would be taxed in the US if his activity in the US rose to the level of a “permanent establishment” (as defined under the treaty), but only to the extent attributable to that permanent establishment. “Selling services to the US” is not a sufficient description to be able to tell if there is a “permanent establishment.”
If he is subject to US tax on a portion of his income as well as UK tax, the UK may (or may not) make some allowance to enable him to avoid double taxation — such as our foreign tax credit or foreign earned income exclusion; whether there are such allowances, however, is something that I cannot tell you, and for which you would need someone more familiar with UK law.