In the past few months, around sixty children have died from the disease in the remote district where malnutrition cases are also soaring.
Since reports of the extent of outbreak filtered out of Papua earlier this month, Indonesia's government has moved quickly to attend to the situation, but is copping criticism for neglecting the health of Papuans over many years.
Papua's police commander, Boy Rafli Amar last week said the number affected by malnutrition had surpassed 10,000, which was exacerbating the measles problem.
An integrated health team to respond to the crisis has been dispatched from the Papua Province Health Office, Indonesian National Army, and the Ministry of Health.
The team was armed to supply Asmat villagers with medicine, vaccines, medical equipment and nutritious food.
A spokesman for the team told the Antara news agency that they had successfully attended to 117 Asmat villages where he said the measles outbreak had now been suppressed.
The epidemic is being attributed by health officials to food shortage, a poor sanitary environment, and a lack of medical personnel and facilities in Asmat.
Based on information received on Jan 25, a total of 12,398 children in Asmat have been offered medical check-ups and treatment.
Of the total children, 646 are confirmed to be infected with measles.
Despite Papua region's abundant resources, which provide significant revenue for the Indonesian state, Papua continues to lag in human development outcomes.
Across the whole Indonesian republic, the highest poverty rates, in relative terms, are all in its far east provinces of Papua, West Papua and Maluku, according to Jakarta's Statistics Agency.
The health situation is of particular concern. Papua has the lowest life expectancy in Indonesia and the highest infant, child and maternal mortality rates. Diseases such as malaria, leprosy and malnutrition have strong footholds in in Papua, as does HIV/AIDS.
Although since coming to power in late 2014, President Joko Widodo promised to bring greater economic and social development to Papua, as well as improved health care, the welfare of Papuans appears to have deteriorated.
In the past couple of years, several reported outbreaks of endemic diseases in various parts of Papua have killed hundreds of people.
The government still significantly restricts access to Papua for international humanitarian and health NGOs who could help bolster public health services.
The United Nations special rapporteur on the right to health, Dainius Pūras, reported after visiting last year that he was "concerned about the health status of ethnic Papuans" and called for greater health investment.
The NGO Human Rights Watch has fingered blame for the measles outbreak in Papua on the Indonesian government.
It says while Jakarta blames the deaths in remote Asmat regency on nomadic lifestyles, it is the government who has neglected basic health rights of Papuans.
Human Rights Watch said that Indonesia could have prevented the deaths by establishing an effective vaccination programme in Papua. The current programme is patchy and has not included Asmat children, according to the NGO.