How to copy depth buffer to a texture on the GPU?

I want to get the current depth buffer to a texture, to access it in a shader. For various reasons I can't do a separate depth pass, but would need to copy the already-rendered depth.

glReadPixels would involve the CPU and potentially kill performance, and as far as I know glBlitFramebuffer can't blit depth-to-color, only depth-to-depth.

How to do this on the GPU?