  We measured the low-temperature thermal conductivity of a new layered
superconductor with quasi-one-dimensional characteristics, the ternary
telluride Ta$_4$Pd$_3$Te$_{16}$ with transition temperature $T_c \approx$ 4.3
K. The significant residual linear term of thermal conductivity in zero
magnetic field and its rapid field dependence provide evidences for nodes in
the superconducting gap. By measuring resistivity under pressures, we reveal a
superconducting dome in the temperature-pressure phase diagram. The existence
of gap nodes and superconducting dome suggests unconventional superconductivity
in Ta$_4$Pd$_3$Te$_{16}$, which may relate to a charge-density wave instability
in this low-dimensional compound.
