We report the detection of a magnetic field on the Of? p star HD 108. Spectropolarimetric observations conducted in 2007, 2008 and 2009, respectively, with NARVAL@ Telescope Bernard Lyot (TBL) and Echelle SpectroPolarimetric Device for the Observation of Stars at Canada-France-Hawaii Telescope (ESPaDOnS@CFHT) reveal a clear Zeeman signature in the average Stokes V profile, stable on time-scales of days to months and slowly increasing in amplitude on time-scales of years. We speculate that this time-scale is the same as that on which Ha emission is varying and is equal to the rotation period of the star. The corresponding longitudinal magnetic field, measured during each of the three seasons, increases slowly from 100 to 150 G, implying that the polar strength of the putatively dipolar large-scale magnetic field of HD 108 is at least 0.5 kG and most likely of the order of 1-2 kG. The stellar and wind properties are derived through a quantitative spectroscopic analysis with the code CMFGEN. The effective temperature is difficult to constrain because of the unusually strong He lambda lambda 4471, 5876 lines. Values in the range of 33 000-37 000K are preferred. A mass-loss rate of about 10(-7) M(circle dot) yr(-1) (with a clumping factor f = 0.01) and a wind terminal velocity of 2000 km s(-1) are derived. The wind confinement parameter. eta(star) is larger than 100, implying that the wind of HD 108 is magnetically confined. Stochastic short-term variability is observed in the wind-sensitive lines but not in the photospheric lines, excluding the presence of pulsations. Material infall in the confined wind is the most likely origin for lines formed in the inner wind. Wind clumping also probably causes part of the Ha variability. The projected rotational velocity of HD 108 is lower than 50 km s-1, consistent with the spectroscopic and photometric variation time-scales of a few decades. Overall, HD 108 is very similar to the magnetic O star HD 191612 except for an even slower rotation.