Can a plane with air-breathing engines in level flight achieve enough velocity to make a single orbit? A rocket uses engines that carry both fuel and oxidizer. Most are composed of multiple stages and lift a payload to orbit. They are able to burn their engines anywhere in or out of Earth's atmosphere since they carry both fuel and oxidizer. This is necessary to achieve orbital elevation and velocity.

Can a plane with air-breathing engines achieve a high enough speed in level flight such that it reaches orbital velocity? The idea being the plane could use the fuel in its tanks and the air in the atmosphere (at whatever altitude makes the most sense) to gain enough kinetic energy that the plane could then "climb" to an orbit. This would be similar to a zoom climb, but resulting in the plane entering a stable orbit. The plane could burn its engines until flameout as it ascended through the atmosphere. At that time it'd be dependent on kinetic energy alone. The engines, fuel tanks, and other equipment for air breathing flight can be jettisoned at this point if need be.

For the purposes of this discussion, orbit is defined as a trajectory having a minimum altitude of 100 kilometers. Once at this altitude, the vehicle would need to be able to make at least trip around the Earth without requiring additional delta-v from any onboard source. If the path taken was highly eccentric or even whatever it's irrelevant to this discussion as long as the minimum altitude is maintained. There is no requirement for re-entry at a specific location or even that re-entry is survivable by the craft.

The craft could use esoteric materials, fuel sources, engines, and even construction so long as those have been used in at least one prior sub-orbital flight.

If the plane needed a Reaction Control System to control its trajectory once it reaches high altitude this would be acceptable. These systems don't have enough thrust to actually send the plane into 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).