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Undeterred by Berlin’s ever-growing prestige as Germany’s political and cultural centre, Stuttgart continues to ply its trade as a grand patron of the arts. Opening this month is the southern city’s latest cultural attraction, a gallery of modern art.

The £48m gallery can be taken as a sequel to James Stirling, Michael Wilford and Associate’s two show-stopping contributions to Germany’s third largest urban centre: a state art gallery and music school, which were completed in 1984 and 1995 respectively. Yet it shares neither location, architectural appointment nor style with its precursors. What’s more, it has neatly healed a particularly gruesome urban gash inflicted in the 1960s.

Far from being a flamboyant assembly of sculptural stonework like the state gallery, this art box is a pure cube of clear glass. The design, by Hascher Jehle Architektur of Berlin, won an international competition that attracted 340 entries.

The one feature that the glass cube does have in common with the state gallery is the manner in which its approach walkways are arranged to celebrate the surrounding cityscape – albeit in a different way. The glass outer skin encloses not the galleries themselves but a sequence of wide public terraces and stairs that spiral up around them on five floors. As with the external escalator at the Pompidou Centre in Paris, this generous public access route draws visitors off the pavement and lifts them up and around the building. On arrival at the top-floor terrace, visitors are rewarded with a spectacular panorama over the Schlossplatz public square.

In the evening, the whole glass cube becomes a glowing light sculpture and a means of advertising the gallery without actually presenting the paintings themselves. These are hidden from the harmful effects of daylight – as well as the gaze of passers-by – by an internal box of solid walls faced in stone.

Although it is within the historic city centre, the art gallery’s site suffered severe municipal vandalism in the 1960s. It was redeveloped as a wide deck covering a multi-lane underpass and transport interchange that spewed out a constant stream of cars, lorries and tramcars into the Schlossplatz.

The traffic underpass has now been closed and the space used to create two levels of ancillary spaces behind and below the gallery building. On either side of the glass cube, wide and stately flights of stone steps lead people from the upper deck down to the newly pedestrianised Schlossplatz. And beyond one of these open flights of steps, the grand classical stone portico of the former stock exchange stands as if in counterpoint to the glass cube.

Stuttgart has lost a traffic whirlpool and gained a glowing work of modern architecture, an extended piece of Urban Design for the benefit of city strollers and a sizeable gallery of modern art. Not even a highway engineer could object to all of that. Probably.