Uniform cost-distance Steiner trees minimize the sum of the total length and weighted path lengths from a dedicated root to the other terminals. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks. They are a special case of general cost-distance Steiner trees, where different distance functions are used for total length and path lengths. We improve the best published approximation factor for the uniform cost-distance Steiner tree problem from 2.39 to 2.05. If we can approximate the minimum-length Steiner tree problem arbitrarily well, our algorithm achieves an approximation factor arbitrarily close to $ 1 + \frac{1}{\sqrt{2}} $. This bound is tight in the following sense. We also prove the gap $ 1 + \frac{1}{\sqrt{2}} $ between optimum solutions and the lower bound which we and all previous approximation algorithms for this problem use. Similarly to previous approaches, we start with an approximate minimum-length Steiner tree and split it into subtrees that are later re-connected. To improve the approximation factor, we split it into components more carefully, taking the cost structure into account, and we significantly enhance the analysis.