We studied 14 groups of French pottery fragments dated between the 4th and 16th centuries. The potsherds were analyzed using the Thellier and Thellier [1959] method, revised by Coe [1967]. Intensity values were corrected for thermoremanent magnetization (TRM) anisotropy and cooling rate dependence of TRM acquisition. We first analyzed modern ceramics produced following ancient techniques and fired in a wood-burning kiln inside of which field intensity was measured. The recovered mean intensity is within $3% of the expected value, which proves the reliability of our experimental procedure. Thermal experiments carried out at rapid and slow cooling rates clearly indicate that the cooling rate correction is critical in archeointensity studies. Our data indicate that large variations in intensity occurred in France over the last 2000 years. Two relative maxima in intensity are observed, one between the 8th and 10th centuries and the second between the 14th and 15th centuries. Similarities are observed between the archeointensity data from France and Ukraine, yielding some evidence for eastward drift of geomagnetic sources between western and eastern Europe from A.D. 800 to A.D. 1700. We also show that the dipole moment evolution proposed by McElhinny and Senanayake [1982] and Yang et al. [2000] for the last two millennia is likely biased toward higher values, mainly because of the absence of correction for the cooling rate dependence of TRM acquisition in most published archeointensity studies. We finally underline a possible relationship, valid at least in western Europe, between changes in direction and intensity of the geomagnetic field.