Construction must be completed by the start of the games next year – a period of fewer than 500 days. International concern has been expressed that Greece will not have the facilities ready in time.
The scheme is one of 235 projects, ranging from state-of-the-art stadiums to the New Acropolis Museum, designed by Bernard Tschumi, a Swiss architect with an international reputation. Two underground lines will also be constructed.
But Calatrava's project will be the most spectacular of them all.
It will involve erecting two enormous steel arches, each with 304 m span, over the open-air stadium that held the 1896 Olympics. A sunscreen roof in laminated glass will be suspended from them, supported from steel cables.
The roof project was nearly dropped this year when the Olympics committee became increasingly concerned that the project would be half-completed by the time the games opened.
A technical report commissioned by a Greek opposition party also claimed that, in an area prone to earthquakes, the slender arches would suffer from "significant overstress".
Calatrava was able to guarantee the safety of the project at an inspection by the Olympics committee this month.
The architect said: "The new roofs will be supported at four points so that we can prepare the structures with few interventions."
Jacques Rogge, president of the Olympic committee, inspected the work in progress on the construction programme this month. He expressed cautious optimism that the sports facilities would be ready. However he added: "In the first 18 months, frankly speaking, it was too disorganised."