The Burrows Wheeler transform has applications in data compression as well as full text indexing. Despite its important applications and various existing algorithmic approaches the construction of the transform for large data sets is still challenging. In this paper we present a new semi external memory algorithm for constructing the Burrows Wheeler transform. It is capable of constructing the transform for an input text of length $n$ over a finite alphabet in time $O(n\log^2\log n)$ on average, if sufficient internal memory is available to hold a fixed fraction of the input text. In the worst case the run-time is $O(n\log n \log\log n)$. The amount of space used by the algorithm in external memory is $O(n)$ bits. Based on the serial version we also present a shared memory parallel algorithm running in time $O(\frac{n}{p}\max\{\log^2\log n+\log p\})$ on average when $p$ processors are available.