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A (δ, g)-cage is a regular graph of degree δ and girth g with the least possible number of vertices. It was proved by Fu, Huang and Rodger that every (3, g)-cage is 3-connected. Moreover, the same authors conjectured that all (δ, g)-cages are δ-connected for every δ ≥ 3. As a first step towards the proof of this conjecture, Jiang and Mubayi showed that every (δ, g)-cage with δ ≥ 3 is 3-connected. In this paper, we prove that every (δ, g)-cage with δ ≥ 4 and girth g ≥ 10 is 4-connected, whic