Instead, buy a modern carbon bike. 20 lb. bikes can be found for ~$1,000. I've ridden single speed and IMHO it's a silly choice made by those for whom a geared bike isn't enough of a challenge or ego boost or something. Plus riding it will absolutely hurt your back. Which is kind of a good thing in a way because you can't get stronger without it hurting. But a geared bike will also hurt your back just fine.