This simple song had me baffled for the longest time. I knew there had to be an easy way to play it. It finally clicked last night.

If you want to make a fingerstyle solo out of this (instead of mere accompaniment) the melody is right within one fret of every chord here. All easy reaches.
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X02322

In the X00232 wee small XX0332 hours of the XX0432 morning XX0332
While the X00232 whole wide 2X022X world is fast a- 02000X 02003X X02020 -sleep
You 02000X lie a- X0200X wake and 2X222X think about the X2425X X2424X boy
And 7X677X never ever X4665X think of 07675X counting X0222X X021XX 00202X X0322X sheep

When your X00232 lonely XX0332 heart has learned its XX0432 lesson XX0332
You'd be X00232 his if 55455X only he'd X2425X X2424X call
In the 02200X wee 2X020X small 3X020X hours X2313X of the X5420X X4320X X342XX morning
XXX0XX That's X24(0)XX the 0X000X time X2003X you XXXX0X miss XXX0XX him X020XX most 3X2322 of XX0232 all XX0332 XX0432 XX0332 554405