The modulation of the galactic cosmic radiation over the past 1150 years is investigated using 10Be data from Greenland and the South Pole. For this purpose, we introduce the use of 22-year averages to study the long-term modulation. After allowance for secular changes in the geomagnetic dipole, it is shown that the 22-year mean intensity of the galactic cosmic radiation (GCR) in the vicinity of 1--2 GeV/nucleon returned to approximately the same high level at the widely separated times of the Oort (1050 AD), Spoerer (1420--1540), and the latter portion of the Maunder (1645--1715) periods of low solar activity. In terms of the modulation potential, $phi$, this asymptotic intensity corresponds to a mean residual modulation of ~84 MV. The GCR intensity was significantly less during the Wolf (~1320) and Dalton (1810) minima, and $phi$ ~ 200 MV. The higher temporal resolution data from Greenland shows that there were large 11-year and other fluctuations superimposed upon these high intensities during the Spoerer and Maunder minima (Δ$phi$ ≈ 200--300 MV), indicating the continued presence of a substantial and time-dependent heliomagnetic field. Throughout the Spoerer minimum, the GCR intensity repeatedly returned to a condition of very low modulation, indicating that the cosmic ray spectrum incident on the Earth approached the level of the local interstellar spectrum. These results imply the continued presence of either (or both) (1) the normal cyclic variation of the heliospheric current sheet and/or (2) a cyclic variation of the diffusion coefficients throughout these periods of low solar activity. The data indicate that the modulation (i.e., depression) of the cosmic ray intensity during the instrumental era (1933--present) has been one of the greatest in the past 1150 years. Further, approximately the same low value has been attained on five previous widely separated occasions since 850 AD, and we speculate that the heliospheric magnetic field has reached an asymptotic limit at those times. The 10Be data exhibit a previously unrecognized feature, which we have named the precipitous decrease, in which the 1--2 GeV/nucleon intensity decreased by ~40--45% between 1700 and 1739 corresponding to Δ$phi$ > 500 MV, at a time of low but increasing solar activity. A lower cosmic ray intensity than that attained in 1739 was not observed again until after 1950, at a time of high solar activity. These features and the large 11-year modulation events during the Spoerer and Maunder minima indicate that the long-term variations in the GCR intensity are poorly related to sunspot numberduring epochs of low solar activity. It is shown that there is better agreement between the variations in the 10Be data, and the changes in the open solar magnetic flux predicted by the Schrijver et al. <2002> and Solanki et al. <2002> models based on historic sunspot numbers. In particular, they both exhibit characteristics consistent with the precipitous decrease in the 10Be data, although the amplitudes are smaller than implied by the 10Be data.