Chop blackberries and place in medium saucepan; add any juice from chopping.
Add honey and water.
Simmer on medium heat until reduced to half; about 5 minutes.
Remove juice mixture from heat and place in medium bowl; let cool completely.
Add cream, do not mix.
Pour mixture into paper cups until 3/4 full. Place popsicle sticks into filled cups until the stick is halfway in. Use two clothespins to pin popsicle stick from each side to keep from sinking or falling. Place in freezer and allow to freeze for at least 5 hours.