[en] The construction of two wire chambers for the experiment E831 at Fermilab is reported. Each chamber includes three wire planes - one anode and two orthogonal cathodes - in which the wires operate as independent proportional counters. One of the chambers is rotated with respect to the other, so that four position coordinates may be encoded for a charged particle crossing both chambers. Spatial resolution is determined by the wire pitch: 1 mm for cathodes, 2 mm for anodes. 320 electronic channels are involved in the detection system readout. Global counting rates in excess to 10⁷ events per second have been measured, while the average electron-positron beam intensity may be as high as 3 x 10⁷ events per second. (author)