Fulham have allowed Kamara to leave Craven Cottage having brought in Stefano Okaka from Roma.
"I'm delighted to sign for Celtic and I am very much looking forward to joining up with the manager and my new team-mates," Kamara told the club's official website. "Celtic is a magnificent football club. I will be looking to do my very best to contribute and bring the supporters success and hopefully enjoy a long future with the club."
Celtic have also completed the signing of Slane from Well on a four-year deal.
Well chairman John Boyle was left unimpressed by Slane's attitude after the teenager refused the offer of a new contract with his current deal set to expire in the summer.
Boyle told the club's website: "We have tried and tried to get Paul to sign a long-term contract but, much to our extreme disappointment, he has continually refused.
"We have offered the player improved competitive terms in line with other progressing youngsters at the club but, both personally and through his agent, he has refused.
"Paul has been at the club for six years and we have financed his development and nurtured his talent, culminating in his senior debut at the end of last season.
"We have also stuck by and supported him over those years through both personal issues and injury and felt we were due some degree of loyalty for our efforts.
"Sadly, he has chosen to refuse our offers and move on to Celtic."
Slane made his debut last season and featured in all of Motherwell's six Europa League games last summer, but a calf injury that required two operations has restricted him to two substitute appearances in the Scottish Premier League this term.
The winger was targeted by Celtic youth academy chief Chris McCart, who previously performed the same role at Fir Park, and is unlikely to be in Mowbray's immediate plans.
Celtic 4/11, Draw 10/3, Kilmarnock 15/2