The thrice-weekly services will be hauled by the A4 class locomotive Union of South Africa, which pulled London-Edinburgh expresses between the 1930s and 1960s.
They will operate on Wednesdays, Thursdays and Sundays for six weeks on the line between Edinburgh and Tweedbank, near Galashiels, which re-opens on 6 September after a gap of 46 years.
Adult return tickets start at £40, which includes ScotRail travel to Edinburgh.
However, up to six regular ScotRail trains a day will have to be cancelled to make room for the trains because much of the line is single track.
That would reduce their half-hourly frequency to hourly for parts of the day.
Last month, ScotRail ran its first steam trains for decades to Inverness and Carlisle.
Customer experience director Jacqueline Taggart said: “These steam journeys will give people a chance to appreciate the new Borders Railway from the wonderful setting of a vintage steam train.
“This is a fabulous opportunity to showcase the country’s newest railway line, and in such a romantic style.”
Transport minister Derek Mackay said: “Steam services running on the re-opened Borders Railway really will recapture the golden age of Scottish rail travel.”
VisitScotland chairman Mike Cantlay said: “I’m convinced the addition of steam trains will make our new railway one of Scotland’s most sought-after attractions.”