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Lie's third theorem does not hold for Lie groupoids and Lie algebroids. Inthis article, we show that Lie's third theorem is valid within a specific classof diffeological groupoids that we call `singular Lie groupoids.' To achievethis, we introduce a subcategory of diffeological spaces which we call`quasi-etale.' Singular Lie groupoids are precisely the groupoid objects withinthis category, where the unit space is a manifold.  Our approach involves the construction of a functor that maps singular Liegroupoids to Lie algebroids, extending the classical functor from Lie groupoidsto Lie algebroids. We prove that the \v{S}evera-Weinstein groupoid of analgebroid is an example of a singular Lie groupoid, thereby establishing Lie'sthird theorem in this context.

On the integrability of Lie algebroids by diffeological spaces