He replaces Bishop Philip Boyce, who had served in the role since March of last year.
Bishop Boyce was brought out of retirement to replace Bishop John McAreavey, who stood down after celebrating Mass with paedophile priest, the late Fr Malachy Finegan.
An apostolic administrator is a temporary appointment.
Bishop Boyce’s appointment was also controversial, with one retired priest saying he was “disgusted”.
In 2011 Philip Boyce was criticised for his handling of clerical sex abuse allegations when he was Bishop of Raphoe.
In a letter issued to Dromore’s 90,000 Catholics, Archbishop Martin said it was his aim to “provide as much stability as possible for Dromore” while awaiting a decision from Pope Francis about the appointment of a new permanent bishop.
“I am humbled by this request from Pope Francis and grateful for his confidence in me,” Archbishop Martin wrote.
He said one of his first priorities would be to receive a full briefing on safeguarding issues and to be updated on any statutory investigations.
“I shall, of course, fully cooperate with developments in this regard,” he wrote.
Bishop Boyce said that while he was looking forward to retirement, he would be available to provide pastoral support to the diocese if required.