We develop a numerical method for elliptic interface problems with implicit jumps. To handle the discontinuity, we enrich usual P-1-conforming finite element space by adding extra degrees of freedom on one side of the interface. Next, we define a new bilinear form, which incorporates the implicit jump conditions. We show that the bilinear form is coercive and bounded if the penalty term is sufficiently large. We prove the optimal error estimates in both energy-like norm and L-2-norm. We provide numerical experiments. We observe that our scheme converges with optimal rates, which coincides with our error analysis.