Passengers from at-risk countries will have their temperature taken, complete a risk questionnaire and have contact details recorded.
Mr Hunt said screening at Gatwick and Eurostar terminals would start in the coming week.
The Chief Medical Officer says the risk to the UK is low, but expects a "handful" of cases.
Mr Hunt said it was "genuinely very difficult" to predict an exact number of cases, but said the expected figure for the next three months was not in double figures.
In September, around 1,000 people arrived in the UK from Ebola-affected countries in West Africa.
Screening at Heathrow Terminal 1 will begin on Tuesday, before being extended to other terminals at the airport as well as Gatwick and Eurostar. The Department of Health estimates that 85% of all arrivals to the UK from affected countries will come through Heathrow.
Border Force officers will identify passengers to be screened. Nurses and consultants from Public Health England will carry out the testing.
Anyone with suspected Ebola will be taken to hospital.
Passengers deemed at high-risk due to contact with Ebola patients, but with no symptoms, will be contacted daily by Public Health England.
A spokesman for Heathrow said the welfare of "our passengers and colleagues is always our main priority".
He added: "We would like to reassure passengers that the Government assesses the risk of a traveller contracting Ebola to be low."
There is no direct flight from Liberia, Sierra Leone or Guinea so people could arrive at airports that do not screen passengers.
"Highly visible information" will be in place at all entry points to the UK, Mr Hunt insisted.
Patients with suspected Ebola will be taken to hospital and blood sample will be taken to the Public Health England's specialist laboratory for rapid testing.
If the test is positive then the patient will be transferred to the specialist isolation unit at the Royal Free Hospital in London. It is the centre that cared for the British nurse William Pooley, who contracted Ebola in West Africa.
Hospitals in Newcastle, Liverpool and Sheffield are on standby to offer similar facilities if there is a sudden surge in Ebola cases. At total of 26 isolation beds could be prepared at the four hospitals.
Ebola has killed more than 4,000 people in Liberia, Sierra Leone and Guinea.
Mr Hunt said tackling the outbreak in Africa was the "single most important way" of preventing Ebola arriving in the UK.
He added: "We should remember that the international community has shown that if we act decisively we can defeat serious new infectious disease threats such as Sars and pandemic flu.
"The situation will get worse before it gets better, but we should not flinch in our resolve to defeat Ebola both for the safety of the British population and as part of our responsibility to some of the poorest countries on the planet."
Â£125m
Pledged by UK to help fight disease
780 British health staff volunteers helping to cope with the crisis
700 Hospital beds supported by UK - tripling Sierra Leone's capacity
750 Military staff to help construct treatment centre and other facilities
100 Beds on board medical ship RFA Argus being deployed to region
The government had been arguing against screening last week, but there was a sudden change in policy.
Mr Hunt said the medical advice had changed and the UK was preparing for the situation deteriorating in West Africa.
He said: "[The chief medical officer] confirms that the public health risk in the UK remains low and measures currently in place, including exit screening in all three affected countries, offer the correct level of protection.
"However whilst the response to global health emergencies should always be proportionate, she also advises the Government to make preparations for a possible increase in the risk level."
Ebola symptoms: what to do in the UK?
Symptoms of Ebola include fever, headache, vomiting, diarrhoea, bleeding - but these are similar to more common infections like flu and some stomach bugs
If you have these symptoms and had contact with an Ebola patient then ring 111 first, do not go directly to A&E or a GP.
If there has been no contact with Ebola then seek help from 111, your GP or A&E if necessary.
The chances of developing Ebola in the UK are low.
Have you been affected by the Ebola outbreak? Are you travelling from affected countries to Heathrow Airport? You can send us your experiences by emailing to haveyoursay@bbc.co.uk.
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Ebola screening will begin at London's Heathrow Airport on Tuesday, Health Secretary Jeremy Hunt says.