[en] A method of obtaining an electron beam in an electron cooler where the transverse temperature is lower than the cathode temperature is presented. The method is based on the fact that the kinetic energy transverse to the magnetic field, mv²_{perpendicularto} /2, of an electron that passes through a negative magnetic-field gradient is reduced by a factor equal to the ratio between the initial and final field strengths, provided that the field change is adiabatic with respect to the cyclotron motion of the electrons. The requirements on the magnetic field and the effects on cooling forces are discussed. It is concluded that a reduction in transverse electron temperature by a factor of 10 to 120 K is realistic for an electron cooler such as the one in CRYRING. (orig.)