If you used read.table() (or one of it's ilk, e.g. read.csv()) then the easy fix is to change the call to:
read.table(file = "foo.txt", row.names = 1, ....)
where .... are the other arguments you needed/used. The row.names argument takes the column number of the data file from which to take the row names. It need not be the first column. See ?read.table for details/info.
If you already have the data in R and can't be bothered to re-read it, or it came from another route, just set the rownames attribute and remove the first variable from the object (assuming obj is your object)
rownames(obj) <- obj[, 1] ## set rownames
obj <- obj[, -1] ## remove the first variable