In network distributed computing, minimum spanning tree (MST) is one of the key problems, and silent self-stabilization one of the most demanding fault-tolerance properties. For this problem and this model, a polynomial-time algorithm with O(log 2 n) memory is known for the state model. This is memory optimal for weights in the classic [1, poly(n)] range (where n is the size of the network). In this paper, we go below this O(log 2 n) memory, using approximation and parametrized complexity.More specifically, our contributions are two-fold. We introduce a second parameter s, which is the space needed to encode a weight, and we design a silent polynomial-time self-stabilizing algorithm, with space O(log n • s). In turn, this allows us to get an approximation algorithm for the problem, with a trade-off between the approximation ratio of the solution and the space used. For polynomial weights, this trade-off goes smoothly from memory O(log n) for an n-approximation, to memory O(log 2 n) for exact solutions, with for example memory O(log n log log n) for a 2-approximation.