You should remove them from your mailing list.
As a marketer, the last thing you want to do is alienate your audience by sending them unsolicited e-mail messages.
Since the passage of the CAN-SPAM act, all commercial bulk-email must offer the ability to opt-out as well. According to just one portion of the CAN-SPAM Act:
It requires that your email give recipients an opt-out method. You must provide a return email address or another Internet-based response mechanism that allows a recipient to ask you not to send future email messages to that email address, and you must honor the requests.
When you receive an opt-out request, the law gives you 10 business days to stop sending email to the requestor's email address. You cannot help another entity send email to that address, or have another entity send email on your behalf to that address. Finally, it's illegal for you to sell or transfer the email addresses of people who choose not to receive your email, even in the form of a mailing list, unless you transfer the addresses so another entity can comply with the law.
Violating this law can bring fines of up to $11,000. Deceptive commercial email also is subject to laws banning false or misleading advertising.
You will be better off removing that customer from your list.