Born in Louisa County, Virginia, Meriwether moved with his parents to Jefferson County, Kentucky, in 1803. He attended the common schools and engaged in fur trading in 1818 near what is now Council Bluffs, Iowa. He later engaged in agricultural pursuits in Jefferson County.
Meriwether studied law, was admitted to the bar, and commenced practice. He was a member of the Kentucky House of Representatives from 1832 to 1845. He was an unsuccessful candidate for election in 1847 to the Thirtieth Congress. He was a delegate to the state constitutional convention in 1849, and was Secretary of State of Kentucky in 1851.
Meriwether was appointed as a Democrat to the United States Senate to fill the vacancy caused by the death of Henry Clay, and served from July 6, 1852 to August 31, 1852, when Archibald Dixon was elected his successor. He was not a candidate for renomination in 1852.
He was appointed by President Franklin Pierce as Governor of the Territory of New Mexico, after the position was turned down by Solon Borland, from 1853 to 1855. He later served again in the Kentucky House of Representatives from 1858 to 1885, and served as speaker in 1859. After this he retired to his plantation near Louisville, Kentucky. He was interred in Cave Hill Cemetery.