The Papua New Guinean-born player was placed on the transfer list by the Rhinos last year with a valuation of £250,000.
Segeyaro had announced he would not return to Headingley to fulfil the remaining two years of his contract with the Rhinos.
CEO Gary Hetherington said: "We have agreed to terms with James Segeyaro to enable him to sign with the Cronulla Sharks.
"The terms of the settlement are strictly confidential but we can report that the club is satisfied with the financial settlement.
"Importantly, the club has been completely vindicated in the position that they adopted in relation to this dispute. The sanctity of the playing contract must always be respected and complied with."
Hetherington thanked the two national governing bodies for their stance and wished the 26-year-old all the best for the future in the NRL.
“We are pleased that both the RFL and the NRL recognise the importance of the playing contract and the responsibilities associated with it," he added.
“We anticipate that the strong stance that our club took in relation to this matter will ultimately benefit the game of rugby league in both countries.
“The whole saga leaves a bitter taste, but I am pleased it has been resolved amicably and I hope James can go on and fulfil his potential in the NRL."