This may help . . .
Given that you have the exact day/month/year of the birth and birth place you can go to
www.1837online.com . . . it is a pay-per-view site . . . but you can get to look at all entries for a given surname/first name in a birth, marriage or death category by the Quarter and Year . . . July is in the September Quarter.
E.g., you search would be Hughes . . . Mary . . . Birth . . . 1881 . . . September
If your person is listed you will see their name and the District where the event was recorded and the Volume Number and Page Number of the registration.
Llangollen was in the District of Corwen so you (hopefully) should see:
Hughes, Mary Elizabeth . . . Corwen . . . Volume Number . . . Page Number
Do not restrict yourself to finding both first names - often only one name was recorded . . . the District will be the determining factor, i.e., there will perhaps be other Hughes, Mary Elizabeth listed
With that information you could then go to
http://www.gro.gov.uk/gro/content/certificates/and order the certificate.
District of Corwen
Created 1st July 1837. Mainly in Merionethshire, but included part of Denbighshire.
Sub-districts : Corwen; Gwyddelwern; Llangollen.
GRO volumes : XXVII (1837-51); 11b (1852-1930).
Bettws-Gwerfil-Goch, Bottegirbach, Cefnpost, Corwen, Gwyddeiwern, Llandrillio, Llangar, Llansaintfraidd-Glyn-Dyfrdwy.
Registers now divided between Denbighshire South and Wrexham districts.