David R. Kemp died of smallpox while confined as a POW at Camp Douglas, Illinois. His name is among the POW's on the bronze tablets at Oak Woods Cemetery in Chicago.
http://www.graveyards.com/IL/Cook/oakwoods/confederate-names/tablet009.jpg

The following entry from Ancestry.com provides some information. (See Jim Huffman's post below to order all of his military service records.)

Name: David Rowe KEMP
Birth: 1832 in Craven Co., North Carolina
Death: 22 Nov 1864 in Camp Douglas, Illinois
Note:
Enlisted on Feb 8, 1862 at Marion, Lauderdale Co, MS. Was a Pvt in Company A 37 MS Inf. Detailed as a color bearer, captured on retreat from Corinth, MS 10-4-1862. In January and February of 1863 he is home sick. On October 31, 1863 he is under arrest for desertion, captured near Kennesaw Mountian, Ga. Claims to have been loyal. Was forced to enlist to avoid Conscript (Draft) and deserted Rebel Army to avail himself of amnesty. Died November 22, 1864 near Camp Douglas of Small Pox.

1860 census Lauderdale Co., Mississippi - page 164 - Beat 5
D.R. Kemp age 28 born NC
Kitty A. age 26 born MS
James C. age 3 born MS
George C. age 1 born MS

Father: John C. KEMP b: 1795 in New Bern, Craven Co., North Carolina
Mother: Anna ROWE b: 13 Mar 1797 in ,Craven Co.,North Carolina

Marriage - Kesiah "Kitty" Ann FINLEY b: Abt 1834 in Mississippi

Children:
James C. KEMP b: Abt 1857 in Mississippi
George C. KEMP b: Abt 1859 in Mississippi
Thomas F. KEMP b: from 1860 to 1863 in Mississippi