The aim of this study was to investigate whether a chamber-type-specific radiation quality correction factor kQ can be determined in order to measure the reference air kerma rate of (60)Co high-dose-rate (HDR) brachytherapy sources with acceptable uncertainty by means of a well-type ionization chamber calibrated for (192)Ir HDR sources. The calibration coefficients of 35 well-type ionization chambers of two different chamber types for radiation fields of (60)Co and (192)Ir HDR brachytherapy sources were determined experimentally. A radiation quality correction factor kQ was determined as the ratio of the calibration coefficients for (60)Co and (192)Ir. The dependence on chamber-to-chamber variations, source-to-source variations, and source strength was investigated. For the PTW Tx33004 (Nucletron source dosimetry system (SDS)) well-type chamber, the type-specific radiation quality correction factor kQ is 1.19. Note that this value is valid for chambers with the serial number, SN ≥ 315 (Nucletron SDS SN ≥ 548) onward only. For the Standard Imaging HDR 1000 Plus well-type chambers, the type-specific correction factor kQ is 1.05. Both kQ values are independent of the source strengths in the complete clinically relevant range. The relative expanded uncertainty (k = 2) of kQ is UkQ = 2.1% for both chamber types. The calibration coefficient of a well-type chamber for radiation fields of (60)Co HDR brachytherapy sources can be calculated from a given calibration coefficient for (192)Ir radiation by using a chamber-type-specific radiation quality correction factor kQ. However, the uncertainty of a (60)Co calibration coefficient calculated via kQ is at least twice as large as that for a direct calibration with a (60)Co source.