Counterterrorism troops retook the area of the Great Mosque of al-Nuri and are continuing to advance through the remaining streets of Mosul's Old City, Iraq's military said in a statement.
"Their fictitious state has fallen", an Iraqi military spokesman, Brigadier General Yahya Rasool, told state TV.
"In a few days our forces will reach Corniche and bring the battle to its conclusion", said Jawdat.
Shortly after the terror group's rise to power in the northern Iraqi city, IS leader Abu Bakr al-Baghdadi used the mosque to deliver a speech in which he proclaimed the establishment of the group's Caliphate.
In Syria, the insurgents' "capital", Raqqa, is almost encircled by a US -backed, Kurdish-led coalition.
The Iraqi authorities expect that the fighting in Mosul will end in the days to come, as the Islamic state's area is shrinking.
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Tillerson called on all parties to lower their rhetoric and to seek unity in the fight against terrorism . However, Tillerson said the proposals provided a basis for dialogue leading to a solution of the crisis.
"The houses are very old", he said, referring to the Old City, "so any bombardment causes them to collapse completely".
Our correspondent says units from the Counter-Terrorism Service (CTS), Emergency Response Division (ERD), Federal Police and Iraqi Army began what they called the "final battle" for the Old City on Thursday.
Earlier in the week, several ISIS commanders were killed at a different location in Iraq after one of the group's suicide bombers blew himself up at a meeting in response to the terrorists' losing ground in Mosul.
A USA -led worldwide coalition is providing air and ground support in the eight-month-old offensive.
The Iraqi government once hoped to take Mosul by the end of 2016, but the fighting has dragged on as the militants reinforced positions in civilian areas, launched suicide auto bomb attacks, laid traps and kept up sniper and mortar fire.
United Kingdom forces have also hit 69 targets in support of Syrian Democratic Forces in Raqqa, where Sir Michael said there is "irreversible momentum and progress" towards the defeat of IS. Hundreds of civilians fleeing the Old City have been killed in the past three weeks.
Iraqi forces on Thursday captured the compound of a landmark mosque in Mosul that was blown up last week by the Islamic State group - a hugely symbolic site from where the top ISIS leader declared an Islamic "caliphate" almost three years ago.