"Our failure to deliver Brexit has put our country and our party in grave peril.
The leadership I offer is based on one simple truth: without Brexit there will be no Conservative government and maybe no Conservative Party," he said.
"If we want a deal we are going to have to engage seriously with Brussels.
From my conversations with European leaders, it is clear to me there is a deal to be done, they want us to come up with proposals."
Hunt, who is foreign secretary, said that with a "more aggressive Russia" and a "more autocratic China", he would boost defence spending if he became prime minister.
"Peace comes through strength," Hunt said.
"With a more aggressive Russia and a more autocratic China I pledge to increase the proportion of our GDP we spend on defence."