Modeling dependence structure of a bivariate survival data is one of the main issues in survival. Copula models and frailty models have become popular for modelling the dependence structure in bivariate survival data. The distribution of the frailty variable is generally assumed to be continuous and, in some cases, when a cure rate is present in survival data may be better to use a discrete distribution for the frailty random variable. Having zero frailty can be interpreted as being immune or cured. In this paper, we propose a frailty-copula approach induced by discrete frailty to model a bivariate survival data in the presence of a proportion of long-term survivors. We consider a mixed Poisson distribution for the frailty variable, where the distribution of the mixture is a weighted Lindley distribution. We use the maximum likelihood method to estimate the model parameters and evaluate their performance through a Monte Carlo simulation study. We illustrate the performance of the proposed model by means of a real tobacco-stained fingers data set which is obtained from a retrospective cohort study on a sample of 143 smokers screened between March 2006 and January 2010 in a 180-bed community hospital in La Chauxde-Fonds, Switzerland.