Given the following context:

I was stationed on the Isle of Grain – a ghastly place but the first line of defence against invasion – and in our battalion of the Suffolks we had a number of professional musicians. So we formed an orchestra and played in the equivalent of the NAAFI during our spare time. I was the principal cello and we were conducted by the bandmaster, one Lieutenant Bonham. The other boys knew that I was longing to conduct and one day when Bonham fell ill with 'flu, they thought "old Barby" – as I was known – should have a go. It was really rather romantic – I was scrubbing the floor in the Officers' Mess when they came and invited me to take over. We did the Light Cavalry overture and Coleridge-Taylor's Petite Suite de Concert but I can't say I recall the rest of the programme.
While in the army, Barbirolli adopted the anglicised form of his first name for the sake of simplicity: "The sergeant-major had great difficulty in reading my name on the roll-call. 'Who is this Guy Vanni?' he used to ask. So I chose John." After demobilisation he reverted to the original form of his name, using it until 1922.On re-entering civilian life, Barbirolli resumed his career as a cellist. His association with Edward Elgar's Cello Concerto began with its première in 1919, when he played as a rank and file member of the London Symphony Orchestra. He was the soloist at another performance of the concerto just over a year later. The Musical Times commented, "Signor Giovanni Barbirolli was not entirely equal to the demands of the solo music, but his playing unquestionably gave a considerable amount of pleasure." At the Three Choirs Festival of 1920 he took part in his first Dream of Gerontius, under Elgar's baton, in the LSO cellos. He joined two newly founded string quartets as cellist: the Kutcher Quartet, led by his former fellow student at Trinity, Samuel Kutcher, and the Music Society Quartet (later called the International Quartet) led by André Mangeot. He also made several early broadcasts with Mangeot's quartet.

answer the following question:

What was the name of the concerto that Signor Giovanni Barbirolli played the cello soloist? 

Edward Elgar's Cello Concerto