The endpoint region of the P-spectrum of tritium was remeasured by an electrostatic spectrometer with magnetic guiding field. It enabled the search for a rest mass of the electron-antineutrino with improved precision. The result is in: = -39 f 34,t,t f lSsysf ( e~/ c~)~, from which an upper limit of m, < 7.2 ev/c2 may be derived. The experiment yields the 'atomic mass difference m (T) -rn ( 3~e ) =-18 591 f 3 ev/c2.The cjuestion of a no*-vanishing neutrino rest mass is of fundamental importance in .particle and astrophysics. It therefore caused considerable excitement when Lubimov and collaborators [1',2] published a finite value for the electron-antineutrino rest mass m, obtained from a measurement of the endpoint region of the P-decay of tritium with a new, dedicated spectrometer of high resolution and luminosity [ 3 ] . It took a number of years until this result was checked with equivalent instrume~lts by groups in Ziirich [4] and Los Alamos [5] and disproved due to essential improvements of source and data analysis [6,7]. At present, it seems that this generation of experiments is capable of reaching a precision of am: = 100 (ev/c2 l 2 (see table 3 ) .A radically different type of P-spectrometer was proposed independently at several places [8,9]. It acts as an electrostatic filter guiding the electrons adiabatically along the,.lines of an inhomogeneous magnetic field. If the source is placed in a strong magnetic field Bo a n d the retarding potential Uo reaches its maximum in a much weaker magnetic field B1, the full forward solid angle of electfons emitted with energy E is analyzed with a filter width of AE = (BI /Bo) E. This is due to the adiabatic transformation of transverse cyclotron energy E l around the B-lines into longitudinal energy Ell along the B-lines by the magnetic field gradient reducing E l (which cannot, be analyzed electrostatically) in proportion to the magnetic field. Based on this principle,~two somewhat different spectrometers have been set up in recent years at INR in Troitzk [ 101 and at Mainz University. For details of the design, function and performance of the Mainz solenoid retarding spectrometer (SRS) and for further references see refs. [ l l, 121.The experiment described here was performed under the follo-wing conditions. The source is placed at a field Bs = 0.96B0,' slightly in front of the field maximum of the source solenoid which is set to Bo = 2.4 T, limiting the accepted polar angles to 19 < 78". The magnetic field reaches its minimum BI = 8 x T in the symmetry plane of the spectrometer, where Uo maximizes. Retardation of the electrons and reacceleration after the filter is provided by two sets of electrodes arranged symmetrically around the central one. Under these conditions the rise of the transmission from 0 to 1 within the interval E (1 -BI / B o ) G eUd d E is given by [12] Elsevier Science Publishers B.V.