  We present a novel method to measure the strength of interstellar magnetic
fields based on ultraviolet (UV) polarization of starlight, which is in part
produced by weakly aligned, small interstellar grains. We begin with
calculating degrees of alignment of small (size $a\sim 0.01\mu$m) and very
small ($a\sim 0.001\mu$m) grains in the interstellar magnetic field due to the
Davis-Greenstein paramagnetic relaxation and resonance paramagnetic relaxation.
We compute the degrees of paramagnetic alignment with the ambient magnetic
field $B$ using Langevin equations. In this paper, we take into account various
processes essential for the dynamics of small grains, including infrared (IR)
emission, electric dipole emission, plasma drag and collisions with neutral and
ionized species. We find that the alignment of small grains is necessary to
reproduce the observed polarization in the UV, although the polarization
arising from these small grains is negligible at the optical and IR
wavelengths. Based on fitting theoretical models to observed extinction and
polarization curves, we find that the best-fit model requires a higher degree
of alignment of small grains for the case with the peak wavelength of
polarization $\lambda_{\max}<0.55\mu$m, which exhibits an excess UV
polarization relative to the Serkowski law, compared to the typical case
$\lambda_{\max}=0.55\mu$m. We interpret the correlation between the systematic
increase of the UV polarization relative to maximum polarization (i.e. of
$p(6\mu m^{-1})/p_{\max}$) with $\lambda_{\max}^{-1}$ by appealing to the
higher degree of alignment of small grains. We identify paramagnetic relaxation
as the cause of the alignment of small grains and utilize the dependence of the
degree of alignment on the magnetic field strength $B$ to suggest a new way to
measure $B$ using the observable parameters $\lambda_{\max}$ and $p(6\mu
m^{-1})/p_{\max}$.[Abridged]
