Combine the water and ginger in a small saucepan. Bring to a simmer over moderate heat, and simmer for 5 minutes. Remove pot from the heat and reserve. In a bowl beat the egg yolks with the sugar until thick and pale, and wide ribbons fall from whisk. Set aside.

In a large, heavy saucepan, combine the milk, cream, and cooked ginger, and bring to a boil.

Slowly add about 1/4 cup of the hot milk mixture to the beaten eggs, whisking as you pour. Immediately scrape the tempered egg mixture into the hot milk mixture, whisking to combine. Cook over moderate heat, whisking slowly, until it reaches custard stage, 180 degrees F. It should be thick enough to coat the back of a wooden spoon. Don't let the custard boil.

Immediately pour hot custard through a sieve pressing down on the ginger to extract all the juices. Discard the ginger. Refrigerate until cool or chill over ice water, stirring occasionally.

Just before freezing, adjust mixture to taste with the lemon juice. Freeze the ice cream in an ice cream maker according to the manufacturer's instructions and serve.