HD 3167 is a bright (V = 8.9 mag) K0 V star observed by NASA’s K2 space mission during its Campaign 8. It has recently been found to host two small transiting planets, namely, HD 3167b, an ultra-short-period (0.96 days) super-Earth, and HD 3167c, a mini-Neptune on a relatively long-period orbit (29.85 days). Here we present an intensive radial velocity (RV) follow-up of HD 3167 performed with the FIES@NOT, HARPS@ESO-3.6 m, and HARPS-N@TNG spectrographs. We revise the system parameters and determine radii, masses, and densities of the two transiting planets by combining the K2 photometry with our spectroscopic data. With a mass of 5.69 ± 0.44 M ⊕, a radius of 1.574 ± 0.054 R ⊕, and a mean density of {8.00}-0.98+1.10 {{g}} {{cm}}-3, HD 3167b joins the small group of ultra-short-period planets known to have rocky terrestrial compositions. HD 3167c has a mass of {8.33}-1.85+1.79 M ⊕ and a radius of {2.740}-0.100+0.106 R ⊕, yielding a mean density of {2.21}-0.53+0.56 {{g}} {{cm}}-3, indicative of a planet with a composition comprising a solid core surrounded by a thick atmospheric envelope. The rather large pressure scale height (∼350 km) and the brightness of the host star make HD 3167c an ideal target for atmospheric characterization via transmission spectroscopy across a broad range of wavelengths. We found evidence of additional signals in the RV measurements but the currently available data set does not allow us to draw any firm conclusions on the origin of the observed variation.