  New CCD photometry is presented for the hot overcontact binary DK Cyg,
together with reasonable explanations for the light and period variations.
Historical light and velocity curves from 1962 to 2012 were simultaneously
analyzed with the Wilson-Devinney (W-D) synthesis code. The brightness
disturbances were satisfactorily modeled by applying a magnetic cool spot on
the primary star. Based on 261 times of minimum light including 116 new timings
and spanning more than 87 yrs, a period study reveals that the orbital period
has varied due to a periodic oscillation superposed on an upward parabola. The
period and semi-amplitude of the modulation are about 78.1 yrs and 0.0037 d,
respectively. This detail is interpreted as a light-travel-time effect due to a
circumbinary companion with a minimum mass of $M_3$=0.065 $M_\odot$, within the
theoretical limit of $\sim$0.07 M$_\odot$ for a brown dwarf star. The observed
period increase at a fractional rate of $+$2.74 $\times $10$^{-10}$ is in
excellent agreement with that calculated from our W-D synthesis. Mass transfer
from the secondary to the primary component is mainly responsible for the
secular period change. We examined the evolutionary status of the DK Cyg system
from the absolute dimensions.
