The Tera Electronvolt Superconducting Linear Accelerator TESLA is the only linear electron-positron collider project based on superconductor technology for particle acceleration. In the first stage with 500 GeV center-ofmass energy an accelerating field of 23.4 MV/m is needed in the superconducting niobium cavities which are operated at a temperature of 2 K and a quality factor Q 0 of 10 . This performance has been reliably achieved in the cavities of the TESLA Test Facility (TTF) accelerator. The upgrade of TESLA to 800 GeV requires accelerating gradients of 35 MV/m. Using an improved cavity treatment by electrolytic polishing it has been possible to raise the gradient to 35 - 43 MV/m in single cell resonators. Here we report on the successful transfer of the electropolishing technique to multi-cell cavities. Presently four nine-cell cavities have achieved 35 MV/m at Q 0 510 , and a fifth cavity could be excited to 39 MV/m. In two high-power tests it could be verified that EP-cavities preserve their excellent performance after welding into the helium cryostat and assembly of the high-power coupler. One cavity has been operated for 1100 hours at the TESLA-800 gradient of 35 MV/m and 57 hours at 36 MV/m without loss in performance