  We present a precise measurement of the proton longitudinal double-spin
asymmetry $A_1^{\rm p}$ and the proton spin-dependent structure function
$g_1^{\rm p}$ at photon virtualities $0.006~({\rm GeV}/c)^2<Q^2 < 1~({\rm
GeV}/c)^2$ in the Bjorken $x$ range of $4 \times 10^{-5} < x < 4 \times
10^{-2}$. The results are based on data collected by the COMPASS Collaboration
at CERN using muon beam energies of $160~{\rm GeV}$ and $200~{\rm GeV}$. The
statistical precision is more than tenfold better than that of the previous
measurement in this region. In the whole range of $x$, the measured values of
$A_1^{\rm p}$ and $g_1^{\rm p}$ are found to be positive. It is for the first
time that spin effects are found at such low values of $x$.
