I checked Federal rolls and did not find him under either spelling.

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Ambrose Fair, age 20/29?, h Capt. Moore's Company, Adams' Regiment, Arkansas Infantry, 2nd Brigade, Arkansas Volunteers,* mustered in and enrolled August 11, 1862 at King R. Township, Madison, County, Ark. by H. M. Moore for the duration of the war, no further records

* This Regiment was designated at various times as Peel's Regiment Arkansas Infantry, Fitzwilliams' Regiment Arkansas Infantry. During the same time it was also known as 3rd Infantry Infantry, 3rd Brigade Arkansas Volunteers.

The Regiment was disbanded after the battle of Prairie Grove, Ark., the officers mustered out and the men distributed among other organizations in General Cabell's command.

M317: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Arkansas

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O.R.-- SERIES I--VOLUME XXII/1 [S# 32]

DECEMBER 7, 1862.--Battle of Prairie Grove, Fayetteville, or Illinois Creek, Ark.
No. 36.--Reports of Maj. Gen. Thomas C. Hindman, C. S. Army, commanding First Corps, Trans-Mississippi Army, including preliminary skirmishes.

HEADQUARTERS FIRST CORPS, TRANS-MISSISSIPPI ARMY,
Camp near Fort Smith, Ark., December 25, 1862.

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Of all the troops engaged on our side, Adams' Arkansas regiment alone dishonored itself. It was well armed, ably commanded, and surrounded by good soldiers from the same State, setting it an example of courage and patriotism; but, after delivering a single fire, the greater part of the men broke ranks, threw down their arms, and shamefully fled, many of them even deserting to the enemy. The field and staff officers who had been appointed rallied about 75 around the colors, and these did much to redeem the reputation of the regiment. With but few exceptions, the company officers exerted no influence. The other troops displayed the greatest courage, constancy, and enthusiasm. There was no place of shelter upon any portion of the field. Wounds were given and deaths inflicted by the enemy's artillery in the ranks of the reserves as well as in the front rank. During five hours, shell, solid shot, grape and canister, and storms of bullets swept the entire ground. Many gallant officers, and many soldiers equally brave, fell dead or wounded, but their comrades stood as firm as iron. Volunteers maintained their reputation. Conscripts rose at once to the same standard, and splendidly refuted the slanders put upon them by the class of exempts.
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T. G. HINDMAN,
Major-general, Commanding.
p. 143