Recent work by McCracken <2001> shows that 10Be production rates by cosmic rays on the polar plateau are little affected by geomagnetic field changes in the last few hundred years. Also, the 10Be observed in ice cores on the polar plateau probably originated at high latitudes and precipitated to the Earth in about 1 year, according to McCracken. As a result of this assumption, ice core records of 10Be concentration extending back several hundred years, including the Maunder minimum, have the potential to study the solar modulation of cosmic rays on a time scale extending back several hundred years. These ice core records indicate that the 10Be concentration at the time of the Maunder minimum was ~2.0 times what it was during recent sunspot minima in 1965 and 1976. We have examined 10Be production in the atmosphere using new data related to the interstellar cosmic ray spectrum and the effects of solar modulation as determined from Voyager spacecraft data in the outer heliosphere. We have used the FLUKA Monte Carlo program along with new cross-section data to calculate the production of nucleons and 10Be nuclei in the atmosphere. These calculations show that 10Be temporal variations are sensitive indicators of low-energy solar modulation. Our calculations of 10Be production are able to reproduce well the factor ~1.5--2.0 change in 10Be observed in the ice core data as a result of the 11-year solar modulation. We are also able to show that starting as recently as the sunspot minimum of 1954, the cosmic ray intensity at the Earth was higher than it was during more recent minima. The cosmic ray intensity during these minima time periods represents the residual modulation between the Earth and interstellar space. The 10Be measurements are consistant with the fact that given the interstellar cosmic ray spectrum used in this analysis, this residual modulation was small or zero at the time of the Maunder minimum.