Can a plane with air-breathing engines in level flight achieve enough velocity to make a single orbit?
No.

Currently the altitude record for a jet is 37,650m (This is a zoom and climb record, not sustained flight).

Ralph J sum up the issue pretty well, but let me add some data:

Ramjets are the kind of engines that looks the closest to what you want to achieve. They top out at 2km/s Orbital speed would be 7km/s. And that's with a minimal payload. 

There is an inextricable issue with forgoing O2: 


You need significant O2 to have thrust
You cannot get thrust if there is atmosphere (Drag+heating)


Lastly; to achieve an orbit, you need a circularisation burn, that can only be done outside of the atmosphere.

The lowest point of your orbit cannot be higher than the lowest burn performed. 
Using only atmospheric O2; your orbit will be bound to reenter on achieving your first revolution.

(If you intend to achieve escape velocity however, this is not an issue anymore).