This article examines the first years, 1926-1940, of the Institut Henri Poincaré (IHP) in Paris with regard to one particular aspect of the modernization of French mathematics, which is called here « concrete, institutionally backed internationalization ». (This aspect of modernization differed from another one, promoted by the contemporary and youthful Bourbaki enterprise, which remains largely outside the picture here.) The IHP, founded by Émile Borel principally with the help of money from the American Rockefeller philanthropies (in particular the International Education Board), had a dual modernizing effect on French mathematics. In addition to encouraging the relation between mathematical physics and the modern discipline of stochastics, it provided a meeting place and institutional center for all Parisian mathematicians and supported communication with mathematicians and physicists from other countries. The article also describes the collaboration of the leading figures at the IHP proper (Borel, Maurice Fréchet, Georges Darmois), of Paul Lévy (École polytechnique) and the generation of their influential students.