Recently, a new low-noise record of solar activity has been reconstructed for the past 9400 years by combining two 10Be records from Greenland and Antarctica with 14C from tree rings This record confirms earlier results, namely, that the Sun has varied with distinct periodicities in the past. We present a prediction of mean solar magnetic activity averaged over 22 years for the next 500 years mainly based on the spectral information derived from the solar activity record of the past. Assuming that the Sun will continue to vary with the same periodicities for the next centuries, we extract the spectral information from the past and apply it to two different methods to predict the future of solar magnetic activity. First, the two methods are tested by predicting past changes. Our methods are able to predict periods of high and low solar activities for a few centuries in the past. However, they are less successful in predicting the correct amplitude. Then, the methods were used to predict the period 2000–2500. Both methods predict a period of low activity around 2100 A.D. Between 2100 and 2350 A.D., the results are inconsistent regarding the duration of the low-activity state in 2100 A.D. and the level of activity until 2250 A.D. Around 2250 A.D., both methods predict a period of moderate activity. After 2350 A.D., both methods point to a period of high activity. The period of high activity will end around 2400 A.D. and will be followed by a period of moderate activity.