We have measured the melting pressure and pressure in the liquid at constant density of ultra-pure 4 He (0.3 ppb of 3 He impurities) with the accuracy of about 0.5 µbar in the temperature range from 10 to 320 mK. Our measurements show that the anomaly on the melting curve below 80 mK which we have recently observed [1] is entirely due to an anomaly in the elastic modulus of Be-Cu from which our pressure gauge is made of. We thus conclude that the melting pressure of 4 He follows the T 4 law due to phonons in the whole temperature range from 10 to 320 mK without any sign of a supersolid transition.Recent experimental results obtained by Kim and Chan [2,3] have revived great interest to the problem of supersolidity which was first discussed almost 40 years ago [4,5,6]. The supersolid state of matter is characterized by the coexistence of crystalline order and superfluidity. In helium crystals, according to Andreev and Lifshitz [4] and Chester [5], quantum delocalization of point defects (most probably -vacancies) might decrease their activation energy to zero. Bose condensation of such defects can lead then to superfluidity in a crystal, that is, supersolidity. During 1970s and 1980s many experimental groups tried to detect this possible supersolid state by various methods, but unsuccessfully (see [7] for a review).