You Do Need power plug adapter while traveling from Saint Helena to Saint Barthélemy.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E and in Saint Helena is G. This is how they look like.

Saint Helena

In Saint Helena, outlets of type G are in use.

Type G

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E