The company has replaced Carillion as the main contractor on the project after the firm entered into compulsory liquidation in January.
Features include extending Platforms 5 and 6, to the east of the station, and 12, to the west, as Waverley prepares for the arrival of Virgin East Coast's new Azuma trains and ScotRail's new class-385s.
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With platform 12 entering service last December, Story will now complete the work to extend Platforms 5 and 6 to 275 metres.
Kris Kinnear, Network Rail's deputy regional director for infrastructure projects in Scotland, said: "The electrification of our railway will allow us to run faster, longer, greener trains and cut journey times while increasing the number of seats available for passengers on new state-of-the-art train fleets.
"We will be working closely with Story to minimise the delay to this project caused by Carillion's liquidation and to reduce any inconvenience for customers at Waverley as much as possible."
(LM)
