The spin-wave stiffness of a ferromagnetic microwire in the form of an amorphous Fe_77.5Si_5.5B₁₅ strand with a diameter of 10 μm in a glass shell has been measured by the method of small-angle scattering of polarized neutrons. According to the quadratic dispersion relation for the ferromagnet, neutron scattering by spin waves is concentrated inside a cone with a cutoff angle θ_C. The cutoff angle θ_C has been determined by comparing the antisymmetric contribution to the intensity of scattering of polarized neutrons and a model function for a magnetic field H varied from 3.40 to 41 mT. The cutoff angle θ_C decreases according to the formula θC2(H)=θC2−(gμBH+Δ)θ0/Ei,whereθ0=ℏ2/(2Amn). The stiffness of spin waves and the energy gap in the spectrum of spin waves in the Fe_77.5Si_5.5B₁₅ ferromagnetic microwire at room temperature determined from the field dependence are meV θC2(H)are A =82(3) meV Ų and Δ=0.048(2) meV, respectively. It has been shown that this method for the measurement of the stiffness of spin waves can be successfully used in compact (with low and moderate fluxes) pulsed neutron sources based on accelerators.