Update 2017 March 15th: I realised that this approach doesn't work for all genes, unfortunately. For example, the gene TTN (which is an HGNC approved gene symbol) is associated with 600334, 603689, 604145, 608807, 611705, and 613765 but biomaRt returns an NA.
I was interested in the number of Online Mendelian Inheritance in Man (OMIM) disorders a particular gene was associated with, which in this case was FGFR2. Once again it was biomaRt to the rescue. OMIM is a collection of genes and disorders, and the morbid map refers to the disorders. This post is on looking up the OMIM morbid IDs for FGFR2.