Giants have lost five of their seven Betfred Super League games this season and are second from bottom in the table.
They are at home to champions Leeds Rhinos on Friday and travel to the team below them, Catalans Dragons, on Easter Monday.
Assistant-coach Chris Thorman has been placed in temporary charge and the club say they expect to be “inundated” with applications for the vacant role.
Stone, 51, joined Giants in 2016. After featuring in the middle-eights Qualifiers that season, they finished eighth in Super League last year.
They lost 48-10 at Wigan Warriors last Friday, a week after a 38-6 home drubbing by Hull KR.
A club statement said a “disappointing start to 2018 has led the board to make a change”.
Giants managing director Richard Thewlis said: “It is a very difficult call for any club to make in respect of the head coach, but the board felt that a change was needed and that the change should be made sooner rather than later with much of the season still to run.
“Rick Stone is an outstanding man, character and coach, but the results for us simply haven’t worked out as anyone would have wished for.
“Everyone here will be upset to see Rick leave as he is a very popular man, but with this decision now made we focus on supporting Chris Thorman in the role with the recognition that we will be inundated with offers of names wanting to come in – all of which we will evaluate in the fullness of time.
“Chris understands this, but he also understands that this is an opportunity for him to put himself forward too as he knows he is very highly thought of in our club.”
The statement added: “The club would like to place on record its thanks and gratitude to Rick Stone and wish him every success in the future.”