Training translators for specific purposes is a time and effort consuming process. A translator should have well developed language skills, the awareness of linguistic theory and the theory of translation, discourse knowledge, the apprehension of a great range of extra-linguistic factors such as cultural and historical information. Teaching and training translators-to-be at a higher education institution should address and meet the professional needs of the students. To reach the goal the Department of Foreign Languages of Law Institute, RUDN University (Peoples' Friendship University of Russia) designed and implemented a system of various tasks and exercises that are concentrated on the development of the abilities and skills of lingua-cultural mediators (translators in particular) to carry out the professional activity in the domain of law. The assignments were realized in the teaching practice within the framework of the Degree Program "Translator for Specific Purposes (Law)" The paper analyzes the practical experience and the effectiveness of the use of the developed technology; it presents the findings of the teaching experiment. The authors set the examples of various tasks that are aimed at the improvement of the students' lexical and grammar skills, discourse competences and the knowledge of the cultural and social traditions of the countries which language the learners study. Special attention was paid to the use of information communication technologies (ICT) as the sources of reference and background information and materials. Some tasks enhance the students' abilities to self-study and self-correction; extend their memory capacity. The findings of the experiment training are also presented.