We consider here the one-machine serial batching problem under weighted average completion. This problem is known to be NP-hard and to date no approximation algorithm exists. Batching has wide application in manufacturing, decision management, and scheduling in information technology. In this paper, we give the first approximation algorithm for the problem. The algorithm has an approximation ratio of 2 and is a priority algorithm, which batches jobs in decreasing order of priority. We also give a lower bound of (2+ √6) /4 ≈ 1.1124 on the approximation ratio of any priority algorithm and conjecture that there is a priority algorithm which matches this bound. Adaptive algorithm experiments are used to support the conjecture. An easier problem is the list version of the problem where the order of the jobs is given. We give a new linear time algorithm for the list batching problem.