Slow-slip events, or ‘silent earthquakes’, have recently been discovered in a number of subduction zones including the Nankai trough^1, 2, 3 in Japan, Cascadia^4, 5, and Guerrero⁶ in Mexico, but the depths of these events have been difficult to determine from surface deformation measurements. Although it is assumed that these silent earthquakes are located along the plate megathrust, this has not been proved. Slow slip in some subduction zones is associated with non-volcanic tremor^7, 8, but tremor is difficult to locate and may be distributed over a broad depth range⁹. Except for some events on the San Andreas fault¹⁰, slow-slip events have not yet been associated with high-frequency earthquakes, which are easily located. Here we report on swarms of high-frequency earthquakes that accompany otherwise silent slips on Kīlauea volcano, Hawaii. For the most energetic event, in January 2005, the slow slip began before the increase in seismicity. The temporal evolution of earthquakes is well explained by increased stressing caused by slow slip, implying that the earthquakes are triggered. The earthquakes, located at depths of 7–8 km, constrain the slow slip to be at comparable depths, because they must fall in zones of positive Coulomb stress change. Triggered earthquakes accompanying slow-slip events elsewhere might go undetected if background seismicity rates are low. Detection of such events would help constrain the depth of slow slip, and could lead to a method for quantifying the increased hazard during slow-slip events, because triggered events have the potential to grow into destructive earthquakes.