First Minister Carwyn Jones has urged parents to make sure their children have both doses of the MMR vaccine, as his children had.
It comes as the number of measles cases in the Swansea epidemic rises to 886.
Mr Jones told assembly members the message was getting through about the importance of getting vaccinated.
He said: "People should have their children immunised with both doses of MMR. My children were - I've got no problem in saying that."
Public Health Wales (PHW) said 78 more cases had been reported since last Thursday with 80 hospitalised since the outbreak began last November.
Tests are also taking place to establish whether the epidemic had claimed its first victim.
Gareth Colfer-Williams, 25, found dead at his home in Swansea last Thursday, had measles but suffered other health problems including asthma.
Meanwhile hospital staff are the latest people to be targeted for immunisation against measles.
Clinics are being run from 08:00 BST to 20:00 BST at all health units in the Abertawe Bro Morgannwg health board area - which includes Swansea - to ensure all staff are protected, particularly those in children's units.
Health officials have thanked parents who have arranged MMR vaccinations for their children but warn there is an urgent need for more children to be vaccinated.
Thousands of non-routine MMR vaccines have been given in the last few weeks, including more than 10,000 across Wales in the last week alone.
But PHW says the outbreak is showing no signs of ending.
"The hardest-hit age group is those between 10 and 18 so we would remind those young people and their parents that now is the time to be vaccinated," said Dr Marion Lyons, director of health protection for PHW.
Of the new total of 886 cases, 712 have been in the Abertawe Bro Morgannwg area which covers Swansea, Neath Port Talbot and Bridgend; 99 in the Hywel Dda region covering Ceredigion, Carmarthenshire and Pembrokeshire; and 75 in the Powys health board area.
Ten schools in Swansea and Neath Port Talbot are to be visited this week with the MMR jab being offered to more than 1,700 pupils.
Hywel Dda Health Board has announced plans to offer MMR jabs to all children and young people at secondary schools in Carmarthenshire, Ceredigion and Pembrokeshire from next Monday.
School clinics will also be held at all comprehensive schools in Powys, while the Vale of Glamorgan council is asking schools to help the Cardiff and Vale University Health Board organise sessions.
Single vaccine 'risk'
As the MMR drive continues, parents have been warned against taking their children to have single measles jabs at private clinics.
The British Medical Association (BMA) said people would remain at risk for longer than if they had the MMR vaccine.
Dr Mark Temple, co-chair of the BMA public health medicine committee, told BBC Wales Today: "They don't get the protection as quickly and because you have to have six injections with intervals between them, the period to which they're susceptible to one or other of the diseases is elongated.
"They are therefore at risk and, more importantly, others are at risk from them."
At the assembly, Liberal Democrat AM Peter Black raised the issue of a private company offering single measles vaccines which had a phone number on its website for people in Swansea.
He asked who was responsible for regulating the Children's Immunisation Centre.
Leader of the House Lesley Griffiths said the company was registered with English health regulator the Care Quality Commission, not its Welsh equivalent Healthcare Inspectorate Wales.
"It does appear that the Welsh government can do nothing really to prevent companies offering these services in Wales, but obviously this is something we are looking very closely at," she said.
"I should reiterate that the Welsh government does not support the delivery of a single measles vaccine on the grounds that it doesn't convey protection obviously for mumps and rubella."
The Children's Immunisation Centre declined to comment.