France international Digne completed his medical and signed a five-year contract on Thursday, having undergone an operation on his nose the previous day.
After being presented on the Camp Nou pitch, Digne said: "I am very happy to be a Barca player, it is a great club and it is a great day for me."
Barca had said on their official website on Wednesday that the transfer fee could rise by €4m subject to "individual and team results" and that the 22-year-old's contract will contain a €60m buyout clause.
The deal follows Tuesday's confirmation that France centre-back Samuel Umtiti is set to join from Lyon this week for €25m. Umtiti is expected to finalise his move on Friday.
Digne, an unused member of France's squad during this summer's Euro 2016 finals, spent last season on loan at Serie A side Roma. The Italian club reportedly had an option to sign Digne for €16m at the end of his loan spell but decided against making the deal permanent.
However, Lucas will now provide a competititon for long term Jodi Alba at the left back position and that simply mean that Adriano Correia will finally move on from Camp Nou having barely featured over the the last two seasons.
Barcelona has spent close to €50m this summer with play maker Denis Suarez having been the first new arrival of the summer.
Barca are also expected to sign a new right back as the club no longer have a specialist in this position after Dani Alves left for Juventus on a free transfer.
Moreover, Barcelona president Josep Maria Bartomeu had earlier said that the club had €60m available to spend on new signings in addition to any revenue raised through sales.
Lucas Digne is a france international player who signed his first professional contract agreeing a three contract deal with Lille until later where he was promoted permanently to the senior team and assigned the number 3 shirt.
On 17th of July 2013, Digne also signed a five year-deal with Paris- Saint-Germain on a fee believed to be around €15 million.
The french international then later moved on to Roma on a season long deal €2.5 million with an option to make the move permanent at the end of the 2015-2016 season