It follows the first case of the virus diagnosed in a man in the east of the country on Friday.
The HSE said on Monday evening that a second case of the virus has now been confirmed.
Close contacts of the confirmed case are being contacted.
Anyone who is a confirmed case must isolate for three weeks.
The HSE said the case was not unexpected following the presence of monkeypox cases in the UK and many European countries.
It is still waiting for supplies of smallpox vaccine which can be given to close contacts and healthcare workers treating infected cases to arrive here.
A spokeswoman said that for each case, public health is following up those who had close contact with the case while they were infectious.
In order to maintain patient confidentiality, no further information about the cases will be provided, the HSE said. Public health risk assessments have been undertaken, and those who were in contact with the cases are being advised on what to do in the event that they become ill.
The cases in Ireland come after the reporting of more than 200 other confirmed cases of monkeypox in Europe, North America and many other countries worldwide over recent weeks. The vast majority of these cases do not have a travel link to a country where monkeypox is endemic.
Many countries have reported that the cases are predominantly, but not exclusively, in men who self-identify as gay, bisexual or other men who have sex with men (gbMSM).
A multidisciplinary Incident Management Team was established by the HSE when the international alert was first raised and commenced activities to prepare for cases in Ireland. The IMT will continue to actively monitor this evolving international situation.
To assist in Ireland’s response, monkeypox has been made a notifiable disease. This means that medical practitioners and laboratories are required to notify the local Medical Officer of Health/Director of Public Health of monkeypox cases in Ireland.
Monkeypox is a rare disease that is caused by infection with monkeypox virus. The virus is found in some animal populations in remote parts of Central and West Africa, and in the past has caused occasional limited outbreaks in local communities and travellers.
The cases being reported across multiple countries now are unusual because most of the cases do not have a link to travel to these parts of Africa.
There are two types of monkeypox: West African monkeypox and Congo Basin monkeypox. It is the milder, West African, type that is causing the current outbreak.
Monkeypox spreads through close contact, including contact with the skin rash of someone with monkeypox. People who closely interact with someone who is infectious are at greater risk for infection: this includes household members, sexual partners and healthcare workers. The risk of spread within the community in general, is very low.
Symptoms of monkeypox virus infection include itchy rash, fever, headache, muscle aches, backache, swollen lymph nodes, chills and exhaustion.
The rash starts as raised red spots that quickly change into little blisters after around three days. The rash is usually first found on the face, spreading to the mouth palms and soles of the feet, but following sexual contact can also be found in the anogenital areas.
Scabs eventually form before falling off.
Most cases are mild and people recover, although severe illness and death can occur in people with weakened immune systems, pregnant women and very small babies.