Spin-wave-based transmission and processing of information is a promising emerging nanotechnology that can help overcome limitations of traditional electronics based on the transfer of electrical charge. Among the most important challenges for this technology is the implementation of spin-wave devices that can operate without the need for an external bias magnetic field. Here we experimentally demonstrate that this can be achieved using submicrometer wide spin-wave waveguides fabricated from ultrathin films of a low-loss magnetic insulator, yttrium iron garnet (YIG). We show that these waveguides exhibit a highly stable single-domain static magnetic configuration at zero field and support long-range propagation of spin waves with gigahertz frequencies. The experimental results are supported by micromagnetic simulations, which additionally provide information for the optimization of zero-field guiding structures. Our findings create the basis for the development of energy-efficient zero-field spin-wave devices and circuits.