McLaren's wife Bette, daughter Linda Lawson and her husband Alan arrived by car shortly before the service started.
His coffin was carried into the church to the sound of Highland Cathedral, played by piper Cameron Renwick, the nephew of former Scotland centre Jim Renwick.
A public address system was set up to relay the funeral to the hundreds of mourners who could not be accommodated in the church, and who had gathered to pay their respects as the cortege passed through the Borders town after the service.
McLaren's grandson Gregor Lawson told the service that the family had been "blown away" by the tributes, cards, telephone calls and e-mails they had received from across the world since his death.
He said: "We are here to lay to rest a great man - a great Hawick man, a great rugby man and a great family man.
"As a wee laddie he was known as Billy. If he was your teacher, you probably knew him as Mr McLaren. We, his grandchildren, knew him as papa, and he knew us as the Hitler Youth - you didn't think his amazing turn of phrase just went to rugby did you?
"We have been amazed by the response from the media, such touching tributes from the great and the good of the rugby world and far further afield, and just as important people we have met on Hawick High Street over the past week or so.
"So much has already been said by people significantly more important and erudite than me about his unparalleled impartiality, his iconic voice, his professionalism, his gentlemanly nature and his ambassadorship for both rugby and Scotland.
"All I would say is that while we have shed many tears through sadness, a great many have also been shed simply through bursting with pride."
Bill McLaren lived his entire life in Hawick
Mr Lawson said that what had gone into his grandfather's mouth was often even more remarkable that what came out, with marshmallows in soup and red wine with lemonade among the more unusual tastes McLaren had developed over the years.
He paid tribute to staff at Hawick Community Hospital, who looked after McLaren in his final days.
And referring to Bette McLaren, Mr Lawson said: "Papa used to always say he ranked a distant third in nana's affections behind Frank Sinatra and Glenn Miller.
"We all knew that was just a tease - he was actually fourth behind [former Scotland rugby international] Gregor Townsend.
"To my nana, he was a hot water bottle, a Hooverer, a dance partner, her golden boy, the love of her life and soul mate for 62 years.
"Seeing them together last week, it was clear that their love was still as strong and as tender as when they first met 62 years ago."
The funeral service was conducted by Rev Neil Combe, who remembered McLaren as a good neighbour and a very private man who respected the privacy of others.
He added: "One of Bill's greatnesses that will be remembered long after the Voice of Rugby has grown indistinct in people's ears, was his care for his family and for his friends."
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