We present the application of a fast quasi-adiabatic continuous method to the measurement of specific heat at 4 He temperatures, which can be used for the study of a wide range of materials.The technique can be performed in the same configuration used for the relaxation method, as the typical time constants between calorimetric cell and thermal sink at 4.2 K are chosen to be of the order of 30 s. The accuracy in the absolute values have been tested by comparing them to relaxation-method results obtained in the same samples (performed in situ using the same set-up), with a deviation between the absolute values < 3% in the whole temperature range. This new version of the continuous calorimetric method at low temperatures allows us to completely characterize and measure a sample within a few hours with a high density of data points, whereas when employing other methods we typically need a few days. An exhaustive study has beenperformed for reproducibility to be tested. In the present work, we have applied this method to two different substances: CeSb 2 , which exhibits three magnetic transitions at 15.5 K, 11.7 K and 9.5 K, and graphite, both highly-oriented pyrolytic graphite (HOPG) and natural crystals. Our results on these graphites are discussed in comparison with previous published data on different kinds of graphite samples.