In this paper we propose an approximation for the Traveling Tournament Problem which is the problem of designing a schedule for a sports league consisting of a set of teams T such that the total traveling costs of the teams are minimized. It is not allowed for any team to have more than k home-games or k away-games in a row. We propose an algorithm which approximates the optimal solution by a factor of 2 + 2k/n + k/(n − 1) + 3/n + 3/(2 · k) which is not more than 5.875 for any choice of k ≥ 4 and n ≥ 6. This is the first constant factor approximation for k > 3.We furthermore show that this algorithm is also applicable to real-world problems as it produces solutions of high quality in a very short amount of time. It was able to find solutions for a number of well known benchmark instances which are even better than the previously known ones.