The Mono project is your best option. However, it has a lot of pitfalls (like incomplete API support in some areas), and it's legally gray (people like Richard Stallman have derided the use of Mono because of the possibility of Microsoft coming down on Mono by using its patent rights, but that's another story).
Anyway, Apache supports .NET/Mono through a module, but the last time I checked the version supplied with Debian, it gave Perl language support only; I can't say if it's changed since, perhaps someone else can correct me there.