We give two fully dynamic algorithms that maintain a (1+ε)-approximation of the weight M minimum spanning forest (MSF) an n-node graph G with edges weights in [1,W], for any ε>0. (1) Our deterministic algorithm takes O(W2log⁡W/ε3) worst-case update time, which is O(1) if both W and ε are constants. (2) randomized (Monte-Carlo style) works high probability runs O(log⁡W/ε4) time W=O((m⁎)1/6/log2/...