The authors study some properties of a class of transitive flat Lie pseudogroups of infinite type. Their main result is a kind of third fundamental Lie theorem from the theory of Lie groups. A Lie groupoid is a groupoid endowed with a smooth manifold structure (of infinite dimension) such that any isotropy subgroupoid inherits a structure of (infinite-dimensional) Lie group. Then any flat Lie pseudogroup of infinite type can be regarded as a local action of an abstract Lie groupoid.