Combine the chocolate and rum in a heatproof bowl. If using a standard bittersweet or semisweet chocolate (without a percentage on the label), add 1/2 cup of the half-and-half; if using a chocolate labeled 66% to 70% cacao, add 1 cup of the half-and-half.
Set the bowl over a pan of barely simmering water (or set the bowl directly in a wide skillet of barely simmering water); stir frequently until the chocolate is melted and smooth. Note: A high-percentage chocolate may cause the sauce to look curdled. If this happens, whisk in a little more half-and-half or some sugar or both.
To adjust the consistency and sweetness, spoon a little sauce over some ice cream and wait a minute for the sauce to cool before tasting. If the sauce tastes too bitter, stir in some sugar, a little at a time. If the sauce thickens with cooling more than you like, stir in more half-and-half by the tablespoon as needed.
Serve the sauce warm over ice cream, or let cool, cover, and store it for up to a week in the refrigerator. Reheat the sauce by setting a bowl in the skillet of barely simmering water.