This paper examines the hypothesis of conditional convergence within the fifteen countries of the European Union, which became member states before May 2004, and between the groups of the same fifteen member states of EU and the ten countries that became members with the last enlargement. Basic data input was GDP per capita for all EU countries, proxy variables were savings and depreciation rate. The data consisted of time series for 50 years (1950 – 2000) for EU-15 (old EU countries), while for EU-10 (new member states) the performance in terms of GDP from 1995 to 2007 (predicted values) was analyzed. The presence of beta convergence among EU-15 countries and EU-15 and EU-10 (new EU members) countries was investigated in the first part of the empirical analysis. Starting with graphical analysis, the growth of GDP for different countries during the studied period was compared to the starting level of GDP. If the points in the graph are negatively correlated, then this is a sign of presence of beta convergence. Afterwards, the presence of beta convergence was tested by using the same but formalized approach, the regression analysis. If the partial regression coefficient for GDP p.c. is positive and statistically significant, then the presence of beta convergence among selected group of countries can be confirmed with statistical certainty. In the last part of the empirical analysis the presence of sigma convergence was tested. This type of convergence can be calculated as standard deviation of logarithms of GDP p.c. in the group of countries. This procedure measures the dispersion around determined average. If the dispersion is decreasing, that means that the countries are becoming increasingly similar to each other, in terms of the GDP p.c., and one can confirm the (sigma) convergence. In both samples highly statistically significant beta convergence was confirmed. Furthermore, sigma convergence was discovered and proved. This confirms the hypothesis of convergence among the fifteen countries of the European Union in the period from 1950 to 2000. Additionally, convergence of ten EU newcomers to the average level of standards of living in the fifteen countries of the EU in the years from 1995 to 2007 is also discovered. Both confirm the existence of forces of convergence among the member states of the European Union. One of the main objectives of the European Union is real convergence among the member states. To achieve this goal the EU formed a cohesion policy and backed it up with important structural funds. Those are used to finance projects in less developed member countries, such as improving the infrastructure and educational system and to restructure less perspectives industries. In this paper, the real convergence among old and new member countries was proved, which proves that Europe did not fail to reach one of its basic aims.