The presence of extremely large negative linear compressibilities (NLC) in crystalline silver oxalate was discovered in a recent work by using first principles solid-state calculations. Although the minimum value of the NLC was found for a negative applied hydrostatic pressure, in this work the presence of NLC in this material for positive applied pressures is verified experimentally by means of high-pressure X-ray diffraction experiments performed at room temperature in the Beijing synchrotron radiation facility. The results of this study demonstrate with certainty that the compressibility of silver oxalate along [010] crystallographic direction is negative for applied pressures in the range from 0.0 to 0.85 GPa. Since the measured compressibility decreases largely as the pressure decreases, large negative values of the compressibility are expected for negative applied pressures in agreement with the results found using first principles methods. Furthermore, the analysis of the variation of the lattice parameters of the crystal structure of silver oxalate under pressure in the principal axes reference system revealed that this material also exhibits the largest negative area compressibility phenomenon found so far at zero pressure, the values of the compressibilities along two of the principal axes being - 16.7 and - 20.0 TPa-1. The negative compressibility phenomenon in silver oxalate can be rationalized in terms of a ``chains of rotating parallelograms'' structural model.