Superconducting energy gap in RNi(2)B(2)C (R=Y and Lu) has been investigated using a break-junction tunneling technique. The observed differential conductance dI/dV is well described by the BCS density of states with a small lifetime broadening parameter. We have found small gap values for the local low-T-c phases in addition to the bulk T-c of 15.6-16.6 K. The gap value of 2 Delta = 4.5 meV with T-c = 15 K gives the BCS ratio of 2 Delta/k(B)T(c) = 3.5, which is kept constant down to the observed lowest T-c of 4.6 K. These results give clear evidence for the BCS mechanism of superconductivity for these compounds.