When you have identified a candidate who you believe will be a great addition to your organisation and will contribute greatly to your development, you need to be very direct from the outset. The candidate needs to know how much you value them, how much you want them and what you can do for them.
All of your dealings will need to be professional, slick and without haste. By the time you get to final stages you should have a clear picture of what the candidate wants in terms of salary, career, personal growth, and short and long term future. By this stage you should have given them all the reassurances they need that your organisation can offer this and more. Focus on the reasons why they are leaving their current employer and use this to your advantage. Tailor your offer around all of this. Your offer is not just about offering a job with a certain salary and certain responsibilities, the bigger picture is you are offering them everything they weren't getting at their previous employer, and all the other positive stuff you picked up from the recruitment process.
Whilst you will have a certain budget to work with and salaries have to be realistic, don't play games, offer what you can, after you have considered their salary expectations. Trying a bargaining game gives their current employer the opportunity to re enter the process to try and tempt their talented worker back.
If the candidate makes you aware that they have been offered more money to stay, remind them of all the reasons they wanted to leave their current employer in the first place, and what they told you they liked about your organisation. Remind them that extra money will only satisfy them in the short term, and none of the other issues will go away. Tell them about the great future you see for them with your organisation and all that you are prepared to invest in them. Never get in to a bidding war!!