We present ultra-high resolution (R approximate to 10(6)) observations of lines due to interstellar Na I D, Ca II K, KI and Ti II towards K Vel (HD 81188). Five velocity components are identified in both Na I and Cart, and two in KI; an upper limit is recorded for Ti II. The very high spectral resolution has enabled us to resolve fully the intrinsic interstellar line profiles, and thereby obtain reliable measurements of the velocity dispersion parameters (b-values) for all identified velocity components. By comparing the observed b-values for Na I and KI, which differ in mass by almost a factor of 2, it has been possible to derive the kinetic temperature (Tk) and line-of-sight rms turbulent velocity (v(t)) simultaneously for the two clouds identified in both elements. We obtain rigorous upper limits of 0.46 and 0.33 km s(-1) for v(t) in these two clouds, which are definitely subsonic at the measured temperatures. The three-dimensional turbulent velocity (equivalent to root 3 x v(t)) is also almost certainly subsonic for both components, although it might just exceed the isothermal sound speed if the kinetic temperatures were close to the lower limits permitted by the line-profile analysis.