Determining the magnetic field related to solar spicules is vital for developing adequate models of these plasma jets, which are thought to play a key role in the thermal, dynamic, and magnetic structure of the chromosphere. Here we report on the magnetic properties of off-limb spicules in a very quiet region of the solar atmosphere, as inferred from new spectropolarimetric observations in the He I 10830 Å triplet obtained with the Tenerife Infrared Polarimeter. We have used a novel inversion code for Stokes profiles caused by the joint action of atomic level polarization and the Hanle and Zeeman effects to interpret the observations (HAZEL, from HAnle and ZEeman Light). Magnetic fields as strong as ~50 G were detected in a very localized area of the slit, which could represent a possible lower value of the field strength of organized network spicules.