  Kagom\'{e}-staircase compound Ni$_3$V$_2$O$_8$ is an attractive multiferroic
material exhibiting rich phase diagrams. However, the magnetic properties and
magnetic transitions have been studied only above 1.3 K. In this work, we study
the thermal conductivity ($\kappa$) of Ni$_3$V$_2$O$_8$ single crystals at low
temperatures down to 0.3 K and in magnetic fields up to 14 T. In zero field,
the magnetic transitions from the low-temperature incommensurate (LTI) phase to
the commensurate phase (C) and then to a second commensurate phase (C') yield
anomalies in $\kappa(T)$ curves at $T\rm_{LC}$ = 3.7 K and $T\rm_{CC'}$ = 2.0
K, respectively, which indicates a significant phonon scattering by the
critical spin fluctuations. When the field is applied along the $a$ axis, the
field dependence of $\kappa$ displays four anomalies associated with different
magnetic transitions and reveals an undetected magnetic state at subKelvin
temperatures. In addition, the $\kappa(B)$ curves are found to depend not only
on the history but also on the magnitude of applying field. When the field is
applied along the $b$ axis, a high-field phase locating above the LTI and
high-temperature incommensurate (HTI) phases is revealed.
