The patient, who had travelled to Nigeria where they are believed to have acquired the infection, presented with symptoms at the Vicl, Public Health England (PHE) said.
It is the second ever case of the rare viral infection recorded in the UK, after a resident of Nigeria staying at a naval base in Cornwall was diagnosed last week.
However there is “no UK link” between the two patients, PHE said.
Monkeypox does not spread easily between people and most who contract the infection recover within a few weeks, although severe illness can occur in some people.
Dr Nick Phin, deputy director of the National Infection Service at PHE, said it was “very unusual” to see two cases in this short a period of time.
“We know that in September 2017 Nigeria experienced a large sustained outbreak of monkeypox and since then sporadic cases have continued to be reported,” he said.
“It is likely that monkeypox continues to circulate in Nigeria and could therefore affect travellers who are returning from this part of the world, however, it is very unusual to see two cases in such a relatively short space of time.
“We are working hard to contact individuals, including healthcare workers, that might have come into contact with the individual to provide information and health advice.”
The second individual was transferred to the Royal Liverpool University Hospital after testing positive for the infection.
Dr Mike Beadsworth, clinical director of the tropical and infectious diseases unit at the hospital, said: “The patient is being cared for on our specialist infectious and tropical diseases unit, by highly trained staff who are experienced in dealing with a variety of infectious diseases.
“All necessary precautions are being taken by specialist staff and there is currently no risk to other staff, patients or visitors.”
The first individual is still receiving care at the Royal Free Hospital in London.