Tenant terminating early

Tenancy Types and Management

I have a 12-mth AST with a tenant running from 1/7/09, expiring 30/6/10. Under 'Notice to terminate' it states "If either party wishes to terminate this agreement then the Tenant must give at least two calendar months' notice and the Landlord must give two months' calendar notice and such notice is to be given in writing and not to expire earlier then the end of the twelfth month of the term......".
The tenant has written to state that they will be leaving on April 20th and will not pay any rent after this date. Obviously this is well before the expiry of the 12 months. Are they legally obliged to pay rent until the end of the term? IF they do move out who is responsible for utilities, Council tax etc? What are my rights?