Families of bivariate distributions with weak dependence between their margins are of practical importance, despite being relatively uncommon. These families can be constructed using specific copulas. This paper introduces a new family, KSEW, developed using the bivariate Kimeldorf-Sampson (KS) copula and the extended Weibull (EW) distributions. The KSEW family is designed for applications requiring marginal distributions with low but significant levels of dependence, positioned above the threshold of independence. Key properties of the KSEW family were derived, including product moments and the correlation coefficient. The study also explored the uniform, power, Weibull, Rayleigh, exponential, and Lomax distributions as specific cases within the EW family, identifying the maximum correlation coefficient for each. Additionally, the reliability function in the dependent stress-strength model under the KS framework was developed. The concomitants of order statistics and associated information measures using the KSEW distribution were also investigated. The practical applicability of the KSEW family was demonstrated through analysis of two real-world datasets, highlighting its suitability for scenarios involving weakly dependent random variables.