Polar plumes extending from the Sun into the solar corona have long been seen during eclipses, and can now be studied without this restriction with telescopes and spectrometers on board of spacecraft. Despite the large amount of observational data available on this prominent phenomenon, it is not clear whether plumes contribute substantially to the fast solar-wind streams emanating from coronal holes. An understanding of the processes leading to the formation of bright plumes and the surrounding darker inter-plume regions in coronal holes requires a good knowledge of the physical conditions in plumes and their environment. This investigation aims at measuring the electron densities and temperatures in these regions with the help of radiance ratios of ultraviolet emission lines obtained by SUMER on SOHO. It finds densities of about 7 × 10 7 cm −3 in bright plumes and 1.3 × 10 7 cm −3 in inter-plume lanes at ≈45 Mm above the limb. At this height, the total plume cross-section relative to the size of the coronal hole was found to be less than 8%. The densities drop by a factor of roughly two over the next 80 Mm in height, in lanes a little less than seen in plumes. In this height range, the electron temperatures in plumes are ≈7.5 × 10 5 K and ≈1.13 × 10 6 K in inter-plume regions. The effective ion temperatures, deduced from the line widths, are higher and nearly independent of the altitude in plumes, whereas they increase in inter-plume regions, starting from an even higher level. No systematic dependence of the line-of-sight bulk velocities on the brightness could be found in the coronal-hole plasma.