Those berries sound like the old-fashioned kind that had so much flavor. This is my favorite & you can lessen the quantity to fit your freezer:
SINFULLY SCRUMPTIOUS ICE CREAM
4 cups plus 3/4 cup heavy cream
3/4 cup milk
1 1/2 cup sugar
1/2 tsp salt
8 egg yolks, well beaten
5 cups mashed peaches or strawberries

Scald the 3/4 cup cream with the milk in the top of a double boiler over direct heat. Add the sugar & salt. Pour a small amount of the milk mixture into the beaten egg yolks. Add the yolks to the milk mixture & over simmering water cook until thickened to a custard consistency (coating the back of a spoon.) Remove from the heat & cool. Mix the mashed fruit with the remaining 4 cups cream. Blend with the cooled custard.
If using a manual freezer, fill the container 3/4 full. Pack ice & rock salt around the edges, using 3 parts ice to 1 part salt. When the crank gets too hard to turn & the ingredients have solidified to a soft ice cream stage, remove the dasher & pack down the ice cream.
Repack fresh ice & salt around the sides, wrap in burlap or newspaper & let stand 1 hour. Freeze in cartons. If you have an electric ice cream freezer, follow the manufacturer's directions.~~Rare Collection

Sounds like you might have the Alpine Strawberries, which are small but absolutely packed with flavour! When they bred for size, they forgot about flavour, I reckon.

Now, I've never used an ice cream machine, and I've found that the best way to prevent crystals forming in ice cream is as follows: You make up the mixture, then partially freeze it in trays - until it's slushy. Make sure to cover the tray with plastic wrap - and that the plastic wrap touches the ice cream all over to exclude the air. Then, beat the dickens out of it, and refreeze until slushy, repeating the plastic thing, then beat the dickens out of it again. I usually do this 3 times, but you might be happy with twice. The partial-freezing/beating prevents the cream and the water from separating during the freezing process, and makes for a more fluffy ice cream. It's very important that you exclude all the air because that's what forms the crystals.

OK, so here's a recipe. A punnet, in case you don't know, is the tray you buy the strawberries in. They vary in size a bit, but they hold roughly 2 cups. You can of course add more or less berries according to your personal taste. They'll turn to mush with all the beating, but if you want lumps, stir in a few more on the last beating. And you can use other berries in the same way. (Oh, and I wouldn't dream of adding salt to anything, much less ice cream, so I consider that an unnecessary ingredient!!)

Strawberry Ice Cream
3 large eggs
1 1/2 cups sugar
1/2 teaspoon salt
5 cups milk
300g tin evaporated milk
400g tin sweetened condensed milk
2 tablespoons vanilla
1 punnet crushed strawberries, sweetened with 3 tablespoons sugar

Beat eggs in large mixing bowl. Add sugar and salt, beating until completely blended. Bring 5 cups milk to the boil. Add hot milk slowly to egg mixture, beating constantly. Add evaporated and condensed milk and vanilla. Add crushed fruit with its juice. Let cool in refrigerator or freezer. Freeze according to ice-cream makerÂs instructions.

Ummmm.... and just how much is a "punnet", Daisy?

To quote from the original post:

A punnet, in case you don't know, is the tray you buy the strawberries in. They vary in size a bit, but they hold roughly 2 cups.

Thank you so much for your recipes. I am going to try both recipes, although I don't have enough strawberries for 2 batches. (Truth be told, I don't have enough strawberries for 1 batch. It will be strawberry lite.) I will make one batch with strawberry and one with fresh peaches. Thanks again. Carmellia

Oops! Sorry about that Daisyduck. I guess I sort of "slide" over that part of your recipe.... tsk, tsk on me.

Hope you like them, Carmellia.