Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10.02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These constants are all worst case constants that are artifacts of our proofs. In practice, we believe them to be much smaller. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.