This paper presents experimental results in the development of low‐emittance relativistic electron beams with finite energy spread, which—among other applications—is suitable for suppressing collective instabilities in an electron‐ring accelerator. The measurements were performed with a divided cathode in an electron‐beam gun, one half of which is connected via a resistor with the high‐voltage terminal. The energy difference can be varied up to 100 keV in proportion to the resistance, with equal subcurrents. The beam parts are well separated, and their radial distance is about equal to the radial difference of the corresponding electron closed orbits, such that electron‐ring formation with minimum radial betatron oscillations should be possible. The beam emittance is as small as about 100 mrad cm and the current is 800 A.