Sylvester Stallone -- PUPPY TRAGEDY ... Coyote Kills Family Dog

Sylvester Stallone's family is mourning the loss of a loved one ... their adorable pet poochie Phoebe ... who was found dead after a coyote attack earlier this month, TMZ has learned.

It's a heart-breaking story ... Phoebe went missing a couple of weeks ago -- and immediately the family scoured their Bel Air community in a desperate attempt to find their beloved 4-legged friend.

They even put up posters in their community offering a $10,000 reward for Phoebe's safe return.

But the story took a tragic turn when someone discovered Phoebe's body on the Stallone property ... the victim of a coyote attack.

Stallone's rep tells TMZ ... Phoebe's death has been very difficult for the family.