Work on site for client NHS Lothian is expected to begin next year, for completion in 2012.
The contract was procured under the new Framework Scotland arrangements now being used for healthcare project north of the border.
The other contenders were Laing O'Rourke, Interserve, Morrison and a Robertson/Dawn Health joint venture.
Jackie Sansbury, NHS Lothian's director of strategic planning, said: "We had a difficult choice to make as all of the five principal supply chain partners shortlisted made major efforts.
"But we are confident that BAM will help us deliver a landmark hospital."
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The new hospital will be almost twice the size of the present hospital for sick children, which was built in 1895 and will be linked to, but separate from, the Royal Infirmary of Edinburgh.
Ken McAlpine, construction director of BAM's Scottish operation, claimed: "There are very few projects of this scale and nature, not just in the UK but across the world."
The Edinburgh deal is BAM's second major success under the framework agreement. The first was the ВЈ80m Royal Infirmary in Dumfries.
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