The growth of core Internet routing tables has been such that it is now viewed as an impediment to the continued expansion of the Internet. The main culprit is multi-homing that stems from sites' desire for greater reliability and diversity in connectivity. These locally rational decisions have a global impact on the Internet, and there is currently no mechanism to effectively control them. A number of technical solutions are being pursued, but this paper explores the use of a "market mechanism." It formulates a model that accounts for sites' incentives and the impact their connectivity choices have on the size of routing tables, and introduces a pricing scheme that seeks to better reapportion the resulting costs. The model is solved for two configurations that capture different deployment realizations and stages. They demonstrate the scheme's effectiveness in controlling the growth of Internet routing tables, while improving the welfare of sites and Internet Service Providers.