In 1904 he was awarded a Peccot Foundation Fellowship which was to allow young school teachers to spend a semester in a university developing their skills. Baire spent the semester at the Collège de France where he lectured on the subject of his thesis and had the lectures published the next year. Baire returned to Montpellier where he suffered the first severe attack of illness but after a while the worst of the attack passed and he was able to work again.
Sa 1904 siya ay iginawad isang Peccot Foundation Fellowship kung saan ay upang payagan ang mga batang guro sa paaralan ng gastusin ng isang semestro sa isang unibersidad na pag-unlad ang kanilang kakayahan. Baire nagastos ang semestro sa kolehiyo de France kung saan siya lectured sa ang paksa ng kanyang tesis at nagkaroon ng lektura-publish na ang mga susunod na taon. Baire ibabalik sa Montpellier suffered kung saan siya ang unang matinding atake ng mga sakit ngunit matapos ang isang habang ang mga pinakamasama ng atake lumipas at siya ay makapag-trabaho muli.