Adding the posts page to the permalink url

I have a static home page, and a few other static pages, and selected a page named “blog” for my posts. The posts are indexed there as expected. However, if you click on a post’s title, the single post is displayed at the root URI of the site, for example:

https://mydomain/2009/05/24/my-post-name

I expected that selecting the blog page for the posts, the permalinks would be:

https://mydomain/blog/2009/05/24/my-post-name

I can get this behavior by changing the “Blog address (URL)” setting in the General Settings tab, however, then my other pages get the /blog prefix, and the blog index page becomes /blog/blog.

Anyone know of a plug-in or other way to keep the posts on the posts page?