A novel method for direct, high precision mass measurements of relativistic exotic nuclei has been successfully applied in the storage ring ESR at GSI. The nuclei of interest were produced by projectile fragmentation of 930 MeV/u bismuth ions, separated in-flight by the fragment separator FRS, stored and cooled in the ESR. The mass values have been deduced from the revolution frequencies of the coasting cooled ions. We have measured 104 new mass values with a precision of about 100 keV and a resolving power of 3.5×10⁵ for the neutron-deficient isotopes of the elements 57≤Z≤84. This paper presents the experimental method, the mass evaluation and a table of the experimental mass values.