I\'m in a difficult position: We have a 3rd party enterprise system that exposes a Java-based API. However, we are a 100% .Net oriented development team.
I don't quite understand your statement "the only Java application server supported on our infrastructure is WebSphere 6.1" with respect to the question about how to run other stuff. But no, you don't need a full App Server just to expose a web service.
I think this is a good starting point for you: http://docs.codehaus.org/display/JETTY/J2se6HttpServerSPI