Liam Fox, another senior Brexit supporter, was made International Trade Secretary.
He wrote to the prime minister yesterday: "The outcome of the United Kingdom's referendum has created a new situation which the United Kingdom and the European Union will have to address soon".
Johnson said he was "very humbled, very proud to be offered this chance".
"We have a massive opportunity in this country to make a great success of our new relationship with Europe and with the world", he said.
Boris Johnson leaves Downing Street after being appointed foreign secretary.
The United States congratulated May and said it was confident in her ability to steer Britain through the Brexit negotiations.
May also spoke of the "precious bond" between the nations of the United Kingdom, implicit recognition of the tensions generated by the referendum in which England and Wales chose to quit the European Union, but Scotland and Northern Ireland voted to stay, raising the possibility of a new Scottish vote on independence.
Britain's new prime minister Theresa May is expected to announce further appointments to her cabinet today.
May ditched finance minister George Osborne, Cameron's closest ally, and appointed former foreign secretary Philip Hammond to the job instead, charging him with the task of calming fears over the potential damage of leaving Britain's biggest market.
Hammond said Thursday that there will be no emergency budget to deal with the economic fallout from Brexit.
She vowed to lead a government that worked for all, not just the "privileged few", promising to give people who were "just managing" and "working around the clock" more control over their lives.
Martin Schulz, the European Parliament's president, wrote on Twitter: "Leadership issue settled, now I expect we work quickly to deliver certainty".
Removal vans were seen on Tuesday as Mr Cameron, wife Samantha and children Nancy, Arthur and Florence, packed up at their home since 2010 and prepared to return to a life outside the gates to Downing Street.
He was foreign secretary under David Cameron from 2014 to 2016, having previously served as defence and transport secretaries. He said the government is working hard "to do what we want, which is to give a guarantee to European Union citizens that they will have their rights respected". He gambled, and he lost.
The statement also said Netanyahu spoke to Cameron.
May, the second female British prime minister after Margaret Thatcher, also has said she will not initiate the exit negotiations before the end of the year.
The daughter of a Church of England pastor, she is cricket fan with a sober demeanour who lists her hobbies as cooking and walking.
British Prime Minister David Cameron has left his Downing Street home for the last time as leader of the country.
The proposal aims to increase the European Union's ability "to make (military) commitments beyond its borders and to strengthen security in our partner countries and neighbours", Hollande said in a speech on the eve of France's traditional Bastille Day military parade.