We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (2005). More precisely, denoting by n, m, w max , and w min the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T 0 ≥ w max and multiplicative cooling schedule with factor 1 − 1/ℓ, where ℓ = ω(mn ln(m)), with probability at least 1 − 1/m computes in time O(ℓ(ln ln(ℓ) + ln(T 0 /w min))) a spanning tree with weight at most 1 + κ times the optimum weight, where 1 + κ = (1+o(1)) ln(ℓm) ln(ℓ)−ln(mn ln(m)). Consequently, for any ε > 0, we can choose ℓ in such a way that a (1+ε)-approximation is found in time O((mn ln(n)) 1+1/ε+o(1) (ln ln n+ln(T 0 /w min))) with probability at least 1 − 1/m. In the special case of so-called (1 + ε)separated weights, this algorithm computes an optimal solution (again in time O((mn ln(n)) 1+1/ε+o(1) (ln ln n + ln(T 0 /w min)))), which is a significant speed-up over Wegener's runtime guarantee of O(m 8+8/ε).