About 40 minutes before cooking, start barbecue fire.
Remove game hen necks and giblets; reserve for other uses, if desired.
With poultry shears or a knife, split hens lengthwise along one side of backbone.
Pull hens open; place, skin side up, on a flat surface and press firmly, cracking bones slightly, until hens lie reasonably flat.
Rinse and pat dry.
Sprinkle with pepper.
In a 1 to 2-quart pan, stir jelly and butter over medium-high heat until melted.
Stir in lime juice; set aside.
Place hens, skin side up, on grill 4 to 6 inches above a solid bed of medium-glowing coals.
Cook, turning occasionally, for 20 minutes; then baste with jelly mixture.
Continue to cook, basting and turning occasionally, until meat near thighbone is no longer pink when slashed (about 10 more minutes).