"Peter met with chairman this morning, with both parties agreeing that there was mutual benefit in continuing the present arrangements on a more permanent basis and both keen to put an end to continuing speculation.
"Peter has been granted permission to assist Scotland manager Craig Levein in March."
Houston's time as caretaker started badly, with a goalless draw with Kilmarnock, a 7-1 drubbing at the hands of champions Rangers and a 1-0 defeat by Aberdeen.
However, he followed that disappointing start with wins over Partick Thistle, Hamilton, Hibernian, Falkirk and a draw with St Johnstone.
And their recent good form has lifted United to fourth in the league.
Midfielder Craig Conway said the United squad were behind Houston and wanted him to become their new boss.
Speaking last week, Conway said: "It was Peter who said he didn't want the job, not the players who said we don't want him.
"If he wanted to take the job, he would have the full backing of all the players. We all get on really well with him."
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