The voltage noise and current inhomogeneities in a thin c-oriented YBa₂Cu₃O_7-x film on MgO substrate have been investigated. An unusually sharp peak of low-frequency noise was observed 7 K below T_c with width of less than 1 K and strongly nonmonotonous dependence on external magnetic field. The appearance of this peak is found to be related to a single defect which leads to the formation of a weak link in the bottleneck of the current percolation path. This weak link is latent one; it is undetectable by the standard characterization techniques and has no effect on the integral transport properties of the film. We were able to reveal it using the low - temperature scanning electron microscopy which allowed us to determine the current density distribution across the weak link with resolution of 1 micron. The critical current density of the weak link is found to be j_c(77K) > 5×10⁶ A/cm² which is comparable with the critical current density of high-quality YBaCuO films. Analysis of experimental dependences in terms of the resistively shunted junction model shows that it is the Abrikosov vortex motion resulting in the fluctuations of the phase difference across the weak link that leads to the peak of voltage noise at the temperature at which the current through the weak link is close to its critical current.