Context : Aims : We aim to measure the starspot rotation periods of active stars in the Kepler field as a function of spectral type and to extend reliable rotation measurements from F- , G- , and K-type to M-type stars . Methods : Using the Lomb-Scargle periodogram we searched more than 150 000 stellar light curves for periodic brightness variations . We analyzed periods between 1 and 30 days in eight consecutive Kepler quarters , where 30 days is an estimated maximum for the validity of the PDC_MAP data correction pipeline . We selected stable rotation periods , i.e. , periods that do not vary from the median by more than one day in at least six of the eight quarters . We averaged the periods for each stellar spectral class according to B-V color and compared the results to archival v \sin i data , using stellar radii estimates from the Kepler Input Catalog . Results : We report on the stable starspot rotation periods of 12 151 Kepler stars . We find good agreement between starspot velocities and v \sin i data for all F- , G- and early K-type stars . The 795 M-type stars in our sample have a median rotation period of 15.4 days . We find an excess of M-type stars with periods less than 7.5 days that are potentially fast-rotating and fully convective . Measuring photometric variability in multiple Kepler quarters appears to be a straightforward and reliable way to determine the rotation periods of a large sample of active stars , including late-type stars . Conclusions :