The army's 9th armored division seized Al-Shifaa district, which includes the city's main hospitals, alongside the western bank of the Tigris river, a military statement said.
The fall of Shifaa means the Old City in the eastern half of Mosul is now surrounded by U.S.-backed government forces, deployed north, west, south and east, across the river.
The battle for the Old City is becoming the deadliest in the eight-month old U.S.-backed offensive to capture Mosul, Daesh's de facto capital in Iraq and the largest city the group came to control in the country.
A mine explosion on Monday at the Old City frontline killed two journalists, Stephane Villeneuve from France and Bakhtiar Haddad from Iraq, and wounded two other French reporters, according to foreign ministry and diplomatic sources in Paris.
Aid organisations are expressing alarm at the situation of more than 100,000 civilians, of whom half are children, trapped in its old fragile houses with little food, water and medicine and no electricity.
The International Committee of the Red Cross said on Monday sick and wounded civilians escaping through Daesh lines were dying in "high numbers".
The militants are moving stealthily in the Old City's maze of alleyways and narrow streets, through holes dug between houses, fighting back the advancing troops with sniper and mortar fire, booby traps and suicide bombers.
They have also covered many streets with cloths to obstruct air surveillance, making it difficult for the advancing troops to hit them without a risk to civilians.
The Iraqi army estimates the number of Daesh fighters at no more than 300, down from nearly 6,000 in the city when the battle of Mosul started on Oct. 17.
A U.S.-led international coalition is providing air and ground support.