Since the 2000s, the European Commission has been paying particular attention to the digital development of the European economy and society. Since 2014, the Digital Economy and Society Index (DESI) has been one of the key tools for monitoring and measuring. In 2021, the DESI index's core indicators were aligned with the objectives of the 2030 Digital Agenda, which has four dimensions: human capital, connectivity, integration of digital technology, and digital public services. We sought to determine whether convergence between the Member States could be detected using the third dimension of DESI's annual database. The σ-convergence analysis was used to estimate the reduction over time of the differences in digital technology integration between the Member States, while the β-convergence analysis was used to estimate the rate of catching up with the initial level of development. The σ- and β-convergence were not confirmed. We examined the individual indicators (11) of the integration of digital technology to identify critical areas that need to be addressed in the future to ensure that digital inclusion is as widespread as possible. Our analysis focused on the Member States that joined the EU during the Eastern European enlargements. The performance of all individual indicators is typically below the EU average, and, as the β-convergence analysis shows, the average catching-up rate is not encouraging.