Live At The Newport Jazz Festival (1960)

1. The Story (solos by Phil Guilbeau - tp, Leroy Cooper -  bs, John Hunt - flh, Hank Crawford - as, David Newman - ts, Ray Charles - p)
2. Lil' Darlin' (solos by Ray Charles - p, John Hunt - flh, Phil Guilbeau - tp, David Newman - ts, Hank Crawford - as)
3. Blues Waltz (solos by Phil Guilbeau - tp, Hank Crawford - as, John Hunt - flh, David Newman - ts, Leroy Cooper - bs, Ray Charles - p)
4. Let The Good Times Roll (solo by David Newman - ts)
5. Don't Let The Sun Catch You Crying (solo by John Hunt - flh)
6. Sticks And Stones 
7. My Baby (I Love Her, Yes I Do) (ft. Margie Hendricks)
8. Drown In My Own Tears
9. What'd I Say
10. I Believe To My Soul

The concert took place on 2 July 1960. The film was first released as part of the Boxset Pure Genius: The Complete Atlantic Recordings (1952-1959). 
Atlantic/WEA, 20 September 2005. The bonus item on this DVD is an interview with Ahmet Ertegun by Taylor Hackford.

The Story (#1) erroneously is called Joy Ride in most sources.

The film on the DVD is presented by Willis Conover - as part of the series Jazz USA. The filmed concert is also known as Newport Jazz Festival 1960. The footage has been shelved by Historic Films.
The concert or a part of it was possibly aired by CBS in the World Jazz Series, hosted by Mitch Miller (broadcast on 4 consecutive nights, from July 1 to 4). Voice Of America aired the whole concert as a "public service presentation by the United States Information Service".

The audio of the concert, at least partially, is archived in the Voice of America collection at the Library of Congress.
One tape (RGA 0070; RWD 5745 A2-B1) specifies the following titles (I'm quoting the erroneous orthography of the LoC catalog):

1. The blues [= Blues Waltz]
2. Let the good times roll
3. Don't let the sun catch you cryin'
4. I've been accused [= Sticks And Stones]
5. My baby
6. Drowning [= Drown] in my own tears
7. What'd I say
8. I believe [to my soul]

The other tape (RGA 0069 (RWD 5744 B3) just has:

     9.  Li'l Darlin'

Joy Ride is an incorrect identification - this tune is The Story; it obviously didn't get aired; Lil' Darlin' may have been broadcast later, separately.

Ray Charles, backstage with
Columbia Records' Mitch Miller.
Photo by Allan Morrison.

This source also mentions a documentary Newport '60 in four 25-minute parts, with Muddy Waters, Otis Spann, John Lee Hooker, Ray Charles, Sammy Price, Jimmy Rushing. Wolfgang's Vault shares other parts of the festival's blues program, and has an interesting article and dramatic sound squibs on the riots that took place, and the closing of the festival.

Program pages in the festival's
souvenir brochure.

In a later interview, Annie Ross (see photo) remembered, "Finally we did the Newport Jazz Festival. That was the year of the riots. I think it was Oscar Peterson who couldn't get through the people who were protesting. I forget what they were protesting about. Don't even know. But they said to us, "Don't just do 15 minutes. You can do half an hour," which for us was incredible, because we were always being told, "You have 15 minutes" or 10 minutes, and we were just starting to stretch when we got to that point. So they told us to do a half an hour, which we did. That's where I got a chance to dress with the Raelettes. Ray Charles was on. He was phenomenal. But dressing with the Raelettes was a trip. It was wonderful. They loved our music, and we loved them. We went on and on."

Backstage, chatting with with Lambert, Hendricks and
Ross (from Ebony, Sep. 1960). Photo by Allan Morrison.

Personnel:
Musicians: Phil Guilbeau - trumpet, John Hunt - flugelhorn; Hank Crawford - alto saxophone; David Fathead Newman - tenor saxophone; Leroy Cooper - baritone saxophone; Edgar Willis - bass; Milton Turner - drums. The Raelettes: Gwen Berry, Margie Hendricks, Mae Mosely, Bettye Smith.

Wolfgang's Vault has brought the (whole?) audio part of the concert to the Web. It's not completely congruent with the description of the Voice of America tapes kept at the Library of Congress. Also read this.
Wolfgang's has released a (regrettably incomplete) version of the concert on vinyl: Ray Charles - Newport 1960 (Concert Vault Vinyl Series, 2014, NJF3600702-VL).

All tunes, except for The Story:

Joël Dufour has helped me to get many data in this article right.