Low-energy quasiparticle (QP) excitations in the heavy-fermion superconductor URu\(_2\)Si\(_2\) were investigated by specific-heat \(C(T, H, \phi, \theta)\) measurements of a high-quality single crystal. The occurrence of QP excitations due to the Doppler-shift effect was detected regardless of the field direction in \(C(H)\) of the present clean sample, which is in sharp contrast to a previous report. Furthermore, the polar-angle-dependent \(C(\theta)\) measured under a rotating magnetic field within the ac plane exhibits a shoulder-like anomaly at \(\theta \sim 45\) deg and a sharp dip at \(\theta = 90\) deg (\(H \parallel a\)) in the moderate-field region. These features are supported by theoretical analyses based on microscopic calculations assuming the gap symmetry of \(k_z(k_x+ik_y)\), whose gap structure is characterized by a combination of a horizontal line node at the equator and point nodes at the poles. The present results have settled the previous controversy over the gap structure of URu\(_2\)Si\(_2\) and have authenticated its chiral \(d\)-wave superconductivity.