He told parliament that a British-led appeal this week had prompted offers of 3,850 new troops plus further efforts to redeploy some 2,450 troops from elsewhere in former Yugoslavia.
For its part Britain, already the second-biggest contributor to UN forces after France, was sending a second 900-strong battalion to central Bosnia for an initial period of four months.
Finally, up to a further 4,000 international troops are expected to be deployed in Bosnia by the summer, Rifkind said.
The announcement followed this week's initiative by Britain's ambassador to the UN, Sir David Hannay, to round up contributions from some 20 countries to meet the demand for 10,650 troops made by Lieutenant General Sir Michael Rose, the British commander of UN forces in Bosnia.
British officials said France, the Czech Republic, Argentina, Ukraine, Finland, Sweden, Denmark, Spain and Russia were among the countries to have offered new troops.
They said a politically sensitive offer by Turkey to send 1,000 men had been accepted, despite misgivings because of Ankara's sympathy for Bosnian Moslems.
Rikfind said sending reinforcements was the right response to the diplomatic progress made since a NATO ultimatum last month led to a cease-fire in the Bosnian capital, Sarajevo.
Since then the Bosnian government and Bosnian Croat commanders have signed a truce in central Bosnia.
"There have of course been many disappointments in Bosnia before, and there may be others to come. But if the cease-fires in Sarajevo and central Bosnia hold, they could be the first steps towards the ending of the conflict," Rifkind said.
He said the decision to send 900 more soldiers struck a balance between the UN's manpower demands, the prospects of peace taking hold and the need to ensure that the 2,450 British soldiers already in Bosnia were not asked to do the impossible.
Rifkind said an advance party of soldiers from The Duke of Wellington's Regiment would arrive in the Croatian port of Split later on Thursday.
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