**The only person who could morally keep such a secret are those morally-bound to keep it: a priest bound by the seal of confession and in some cases, a physician or health care provider bound by laws forbidding disclosure.
Even health care providers are bound to disclose in some cases, as explained on the US government web site,
AIDS.gov:
"*HIV DISCLOSURE POLICIES AND PROCEDURES
If your HIV test is positive, the clinic or other testing site will report the results to your state health department. They do this so that public health officials can monitor what’s happening with the HIV epidemic in your city and state. (It’s important for them to know this, because Federal and state funding for HIV/AIDS services is often targeted to areas where the epidemic is strongest.)
Your state health department will then remove all of your personal information (name, address, etc.) from your test results and send the information to the U.S. Centers for Disease Control and Prevention (CDC). CDC is the Federal agency responsible for tracking national public health trends. CDC does not share this information with anyone else, including insurance companies. For more information, see CDC’s Questions about Privacy, Insurance, and Cost.
Many states and some cities have partner-notification laws—meaning that, if you test positive for HIV, you (or your healthcare provider) may be legally obligated to tell your sex or needle-sharing partner(s). In some states, if you are HIV-positive and don’t tell your partner(s), you can be charged with a crime. Some health departments require healthcare providers to report the name of your sex and needle-sharing partner(s) if they know that information—even if you refuse to report that information yourself.
Some states also have laws that require clinic staff to notify a “third party” if they know that person has a significant risk for exposure to HIV from a patient the staff member knows is infected with HIV. This is called duty to warn. The Ryan White HIV/AIDS Program requires that health departments receiving money from the Ryan White program show “good faith” efforts to notify the marriage partners of a patient with HIV/AIDS.
DISCLOSURE POLICIES IN CORRECTIONAL FACILITIES
If you are serving time in a jail or prison, your HIV status may be disclosed legally under the Occupational Safety and Health Administration’s (OSHA) Standard for Occupational Exposure to Bloodborne Pathogens. State or local laws may also require that your HIV status be reported to public health authorities, parole officers spouses, or sexual partners*."
aids.gov/hiv-aids-basics/just-diagnosed-with-hiv-aids/your-legal-rights/legal-disclosure/
This is obviously a case where the moral argument in favor of discretion with someone else’s private information pales in comparison with the duty to protect people from contracting deadly diseases where it is possible.
