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This paper studies the problem of constructing a minimum-weight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today is a protocol with running time O(∧(G) + √n log* n) due to Kutten and Peleg [KP95], where ∧(G) denotes the diameter of the graph G. Peleg and Rubinovich [PR99] have shown that (√n) time is required for constructing MST even on graphs of small diameter, and claimed that their result "establishes the asymptotic near-optimality" of the protocol of [KP95].In this paper we refine this claim, and devise a protocol that constructs the MST in Õ(μ(G = w + √n) rounds, where μ(G,μ) is the MST-radius of the graph. The ratio between the diameter and the MST-radius may be as large as Θ(n), and, consequently, on some inputs our protocol is faster than the protocol of [KP95] by a factor of (√n). Also, on every input, the running time of our protocol is never greater than twice the running time of the protocol of [KP95].As part of our protocol for constructing an MST, we develop a protocol for constructing neighborhood covers with a drastically improved running time. The latter result may be of independent interest.