This is an adaptation from Jane Grigson's Good Things, even simpler than her recipe. I've done it successfully with strawberries, cherries, nectarines, peaches, raspberries (in Denmark, with primitive tools in a holiday house) and even cantaloupe, though that's better suited for sherbet ice.
Fruit that's almost too ripe to eat is ideal for making ice cream. When the strawberries are dirt cheap in the market, usually in late June, I buy them by the kilo ("industrial-grade strawberries") especially for that purpose.
Put the fruit through a sieve (I use a Mouli Vegetable, but a puree sieve or a sturdy normal sieve and a wooden spoon will do) or liquidise it in a blender.
Beat the fruit puree with the sugar until the sugar is dissolved. Taste it. It ought to be substantially sweeter than you like your ice cream, because you're going to add cream and it will taste less sweet when it's cold anyway. Add the juice of half a lemon to strawberries, or a drop of almond essence to cherries, to enhance the flavour. (Nectarines, peaches and apricots can stand either, though not both at the same time; experiment!)
Whip the cream until it stands up in soft waves (not really peaks yet) and gradually add the fruit puree to it. Chill the mixture for about half an hour before putting it in the ice-cream maker. Stir to mix if the fruit and cream seem to have separated.
If you don't have an ice-cream maker, fill a metal box (such as a cake tin) with the mixture, cover with foil and put it in the freezer. Give it a good stir after one hour, scraping any already frozen stuff from the sides and bottom and mixing it in. Repeat at half-hour intervals until there's no liquid mixture left.
It's best when eaten immediately, but you can keep it frozen in a tightly covered box for a few weeks. Put the ice cream in the fridge half an hour before you want it (or before dinner) so it can get soft enough to scoop.