How can I obtain a birth record for a person born in Egremont,
Cumberland, England in 1870 ?

Dear laura68,

You are lucky. Registration of births, marriages, and deaths was
mandatory in England since 1837, so it is more than likely that the
birth of the person you have in mind was officially registered with
the Civil Registration Office of the Civil Registration District
Egremont was part of.

From 1837 to 1930, Egremont belonged to the Civil Registration
District of Whitehaven. The historical register records from the old
District are now kept in Whitehaven. You can obtain the birth
cerificate from them, here is their address:

Whitehaven Register Office
College House
Flat Walks
Whitehaven
Cumbria
CA28 7RW
ENGLAND
Phone: +44(0)1946-852690

- or -

Whitehaven Register Office
75 Lowther Street
Whitehaven
Cumbria 
CA28 7RB
ENGLAND
Phone: +44(0)1946-852690

Birth, Death and Marriage certificates can be ordered (in most cases)
from the offices listed for 7.00 UK pounds. You should make payments
to 'The Superintendent Registrar' unless other instructions are given
with the address. If the register entry is not at the office where you
sent the application, they MAY forward your request to the correct
office. Please enclose a self addressed envelope. Within the UK please
send a stamped SAE. From outside the United Kingdom please add 50p to
the above prices to cover postage.

For methods of payment if you live outside the UK, please refer to
this website with helpful instructions:
http://www.genuki.org.uk/ab/#MakingPayments
Genuki: Researching From Abroad, by Brian Randell

Sources:

Genuki: English and Welsh Register Offices
http://www.genuki.org.uk/big/eng/RegOffice/

IDO: Cumbria Civil Registrer Offices - Whitehaven
http://www.ido.co.uk/detail_new.cfm?id=37278.0&city=Whitehaven

Hope this helps!
Best regards,
Scriptor

Very quick and detailed response.  Thanks !