Det låg ett skimmer över Gustafs dagar = There was a shimmer during the Gustavian days

This was a special Gala on the eve of the wedding of King Carl Gustav to Silvia Sommerlath which was broadcast live on television. ABBA were the only pop act on the show.

It was felt that the song had been especially written for the future Queen Silvia but this was not the case as it had been written and recorded the previous year and performed earlier in January 1976 in Germany for The Best of ABBA (Musikladen Extra) in and in Australia recorded at the same time as the Bandstand TV special The Best of ABBA which was used in ABBA Down Under/ABBA in Australia in March 1976.

On 4 May 2012, Swedish channel, SVT broadcast the programme K Special about opera singer Kjerstin Dellert. One very brief clip in the programme showed her performance of O, min Carl Gustaf where she was accompanied on the accordion by Benny from this 1976 royal Gala.

The programme was repeated on 25 December 1976, Sweden.

When The television show The Story of ABBA was broadcast in 1982, it showed a short clip of the introduction of this performance by one of previous players who says, “Now make room for ABBA and music of this day. There’s absolutely no chance for gloom when they start to sing and play!”  
NOTE: the clip in the VIDEO section showing this introduction is not from The Story of ABBA, but from Japan NHK's satellite channel BS-2 2-hour long ABBA NHK Special from the early 2000s.

The clip of Dancing Queen has been released officially on ABBA - The Definitive Collection.

In an article in Swedish newspaper Göteborgs-Posten dated 3 September 1976, claimed that the BBC had bought the performance of Dancing Queen. However, it didn't say anything about when and where it may have been shown, but the same article says that the Dancing Queen PROMO clip had been seen on Top of the Pops. I don't recall seeing this clip on Top of the Pops - only the PROMO (velvet suits) and the clip from The Best of ABBA in the blue satin suits.

The fact there are no close-ups of Frida in the whole clip

Lyrics available on the Official ABBA site

Archive data from Swedish Media Database:
1976-06-18
Namn
Personer Dellert, Kjerstin (medverkande)
Söderström, Elisabeth , 1927-2009 (medverkande)
Gedda, Nicolai , 1925- (medverkande)
Hagegård, Håkan (medverkande)
Silvia , drottning av Sverige, 1943- (medverkande)
Carl Gustaf , kung av Sverige, 1946- (medverkande)
Grupp/institution ABBA (musiker)
Utgivning
År/datum 1976-06-18
Matrisnummer 60-75 4489
Anmärkningar
Anmärkning Med Kung Carl XVI Gustaf
Exemplar
2 tum T94-0515
Ljud Stereo, färg
Mått 2 tum
Leverantör Gåva SVT
Arkivnummer T94-0515
Digital Betacam T01-0537
Ljud Mono, färg
Leverantör Gåva SVT
Arkivnummer T01-0537
MPEG-1 ZS_T01-0537
Filbeskrivning 1,5 Mbit/s, MPEG-1 layer 2, 48 kHz, 128 kbit/s
Kondition material saknas början saknas. hänvisning till fil xa_tf09-0204
Arkivnummer ZS_T01-0537
MPEG-1 XA_TF09-0204
Ljud mono
Färg färg
Filbeskrivning 1,5 Mbit/s, MPEG-1 layer 2, 48 kHz, 128 kbit/s
Arkivnummer XA_TF09-0204

Thanks to Carl Magnus Palm, Melinda, Jason Swan, Claes Davidsson, Dustin, Jozsef, Ian Cole, Harold Hanlon and Intermezzo Magazine for information.
YouTube links c/o RealMusic Thessaloniki, and Nina Ríos