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 $T_0 \ge w_{\max}$ and multiplicative cooling schedule with factor $1-1/\ell$, where $\ell = ω(mn\ln(m))$, with probability at least $1-1/m$ computes in time $O(\ell (\ln\ln (\ell) + \ln(T_0/w_{\min}) ))$ a spanning tree with weight at most $1+κ$ times the optimum weight, where $1+κ= \frac{(1+o(1))\ln(\ell m)}{\ln(\ell) -\ln (mn\ln (m))}$. Consequently, for any $ε>0$, we can choose $\ell$ 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 $O(m^{8 + 8/ε})$.