Don't move a problem around; you'll still have the problem, and it may infect others in the organization.
Since she is not covered by FMLA, I am assuming that she has less than one (1) year of service. If that's the case, and the company, or you, or whoever hired her, was unaware of her condition at the time she applied/was hired, there is no obstacle that I am aware of that would prohibit you from terminating her.
Her absences have been excessive ("once a week"), and with or without a Doctor's note, she cannot meet her obligations in the workplace to perform to expectations. Such non-performance puts a burden on co-workers (who must pick up the slack), supervisor (who must rearrange the workforce and duties), as well as the company (which may be required to pay unnecessary overtime). That's justification for termination.
You do not mention age, but if she is over 40, she may attempt to file an unlawful termination suit. To prevent that, offer her a severance package (or if you have one, expand it) of two (2) weeks and have her sign a release. (Contact me at my email address, and I can go into it in more detail).