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Superconductivity in the cuprate YBa2Cu3O7(YBCO) persists up to huge magnetic fields (B) up to several tens of Teslas, and sensitive direct current (dc) superconducting quantum interference devices (SQUIDs) can be realized in epitaxially grown YBCO films by using grain boundary Josephson junctions (GBJs). Here we present the realization of high-quality YBCO nanoSQUIDs, patterned by focused ion beam milling. We demonstrate low-noise performance of such a SQUID up to B= 1 T applied parallel to the plane of the SQUID loop at the temperature T= 4.2 K. The GBJs are shunted by a thin Au layer to provide nonhysteretic current voltage characteristics, and the SQUID incorporates a 90 nm wide constriction which is used for on-chip modulation of the magnetic flux through the SQUID loop. The white flux noise of the device increases only slightly from 1.3 μΦ0/(Hz)1/2at B= 0 to 2.3 μΦ0/(Hz)1/2at 1 T. Assuming that a point-like magnetic particle with magnetization in the plane of the SQUID loop is placed directly on top of the constriction and taking into account the geometry of the SQUID, we calculate a spin sensitivity Sμ1/2= 62 μB/(Hz)1/2at B= 0 and 110 μB/(Hz)1/2at 1 T. The demonstration of low noise of such a SQUID in Tesla fields is a decisive step toward utilizing the full potential of ultrasensitive nanoSQUIDs for direct measurements of magnetic hysteresis curves of magnetic nanoparticles and molecular magnets.