There are immense career opportunities for the ECE engineering Students in Public Sector i.e Government Departments. In order to get into any government department one needs to have completed the Engineering Degree. Here are the list of some of the Government Departments that requires ECE Engineering students :-

SAIL
BHEL
BSNL
MTNL
ISRO
Indian Oil Corporation

In order to get into these departments one has to give the written exams conducted by them.
Private Sector

Apart from Public Sector a ECE student also have an opportunity in Private Sector. There are many Coporates and MNC's that require the students.Companies Like

Reliance Communications
TATA Communications
TATA Consultancy Services
Infosys
Wipro
You need to decide which sector you wish to join and make your career

Feel free to comment if you have any doubt
Good luck